When Goku initially raised the multiplier of the Kaioken to x3 and 4, it shredded his internal body to the point any external contact became agonising for him. Fast forward to the battle with Freeza and he's augmenting himself with a x10 multiplier with no expressed discomfort or overexertion. It wasn't perceived as a double-edged gamble like the Kaioken x20 was; interestingly enough, that multiplier didn't appear to tear apart Goku's muscle structure like the x4 did back when fighting Vegeta. I would say that constitutes as him having mitigated the strain as he would have otherwise maimed himself entirely if the mechanics of his first real usage of the technique back during the Saiyan arc was unalterably applied to his Kaioken usage in the Freeza arc. Oh, and for what it's worth, he did hold the Kaioken for multiple pages in the original manga. How this applies to Piccolo is, well, we know he regenerated from his body being shattered into pieces with no substantial power loss noted like with Nail or Cell. Compared with his first display of regeneration where reforming an arm proved tiring for him, this is very impressive and it alludes to him lessening the effort required to regenerate; a principal that could hopefully be applied to Kaioken. Piccolo is more creative with techniques as he was able to formulate the Special Beam Cannon whereas most of Goku's originality stems from him creating variations of techniques he learned.
You're right. Whis didn't make any particular note of Piccolo throughout the tournament. Gohan also seems to be neglected as a prospective student. Why is this? I don't see any rational justification for him neglecting to consider either of these two when he should be aware of what they're capable of. I hold Toriyama and Toei responsible for this inane negligence because if they tried to give some valid recognition of their talents, backgrounds and potential it would signal to the audience that something more should be expected of them in the long-term context like Vegeta received. Both creative groups have no intention of making any commitments to those characters in the foreseeable future. Maybe they feel Gohan and Piccolo wouldn't sell all that well in terms of merchandise and promotional content for the franchise. Goku and Vegeta are the most popular characters of the show.
I doubt it. Not even Vegeta's first in-manga zenkai doubled his power and he was dimensions ahead of the Goku who drank the Choshinsui. For Goku to trade equal blows with Daimao in his prime after being felled by a much older version of the Namekian who admitted to using less than half of his power would suggest Goku gained a substantial power increase that multiplied his original strength, at least for that particular era. Afterwards during the Freeza arc, Krillin and Gohan were both calculated to have surpassed 10,000 by Jeice. Gohan's power wasn't singled out as being considerably greater than Krillin's at the time. It was only through subsequent zenkais that a gap between their powers began to form. Not that any of this should pose a hindrance for Piccolo. One of Dragon Ball's core thematical ideas is overcoming your limits and reaching new heights. Every major character has accomplished this through training and some ingenuity. Piccolo shouldn't be any different.
Piccolo didn't say that; he described training with Goku and Vegeta as being too much, not that it's impossible for him to attain the same level of power. Both Saiyans have the advantage of coming equipped with multiple transformations and God Ki. The power scaling in this series is so warped that Toei's writing staff believed Piccolo to just be comparable with a Base Gohan who has been deteriorating in power for years now. What kind of travesty is that? Piccolo never slackened in his training. Say what you will about the effectiveness of his methods, he didn't retire from training though. I'm also incredulous about the lack of hindsight in Toei. We already saw Piccolo training with an SSJ Goku who was dimensions apart from him in a manga cover. How would this situation be any different? I feel this hearkens back to what was mentioned previously about lack of authorial willingness to invest in Piccolo being the root cause for his lack of participation in the heavy training more than some inherent impotence on the part of the character.
As I said, Goku himself got stronger and strengthen his body, which is why his body no longer comes close to exploding when he uses the Kaioken x2 during the Namek Saga. He also only used Kaioken in short bursts, compared to when he used Kaioken x3 against Vegeta. As for Piccolo, maybe he can maintain the Kaioken better than Goku for the reasons you named and I wouldn't be surprise if that was the case. The problem is I had with your original statement is that you claimed that since Piccolo was a genius he could learn and improved the Kaioken. He is a genius, but he has never been shown learning other people's techniques and improving them, just like Goku has never really created his own techniques despite also being a genius. So we can't just say someone has this certain skill because they're a genius, even if Dragon Ball is a simple show. I also wouldn't say Piccolo is more creative since his techniques are actually quite straightforward compared to some of the stuff Goku pulled like having several different version of the Kamehameha, even if he didn't originally invent it.
Whis seems extremely picky about his students. As I said, he didn't originally want Vegeta, he had to be bride. Goku was the only one he took without any question and it's hinted that the reason why is because he wants Goku to replaced Beerus. And, I hate to say this, Toriyama haven't recognition Piccolo's talents, backgrounds, and potential since he merged with Kami. After that, Piccolo's talents is almost never mentioned outside of him being a great teacher and clever. He became the poster child of skill over power. As for Gohan, the show made it a point to tell us that for all of Gohan's talent, he doesn't have a warrior's heart and that in the end he will always fall short of his father because of it. Maybe Whis sense this and knows that Gohan may start strong with his training, but will tapper out after awhile. And I doubt it's about merchandise since Gohan is number three and Piccolo is almost always in the top ten. So I don't think merchandise is as important as you think.
Given how weak Goku was in Dragon Ball, doubling his power isn't that big of an increase. Remember for all the stuff Goku went through in Dragon Ball, his power level after meeting Raditz was only a little over 400. I believe Gohan was 18,000 after he got his power awoke. And Gohan was singled out as being considerably greater by Vegeta since he thought Gohan was Goku after he got his power unlocked and Vegeta would know Goku would be over 10,000 after he healed from their battle. My point about unlocking power is that in most cases, it isn't as dramatic or game changing like when Gohan got his power unlocked.
If it was possible for Piccolo to catch the Saiyans, I think he would have done it. When have Piccolo back down if something was simply 'too much'? This is the same person who trained with Super Saiyan Goku for three years for the androids and Goku usually train in his base form anyway so Piccolo wouldn't have to worry about Goku transforming and kicking his ass. Not like he ever did since again, he trained with Super Saiyan Goku for three years.
Also, "Toei's writing staff believed Piccolo to just be comparable with a Base Gohan who has been deteriorating in power for years now". That wasn't Toei, that was Toriyama. He's the way who made Piccolo weaker compared to a rusty Gohan and this has always been the case with Gohan. Gohan didn't train for seven years, yet Tien another character who keep training get left in the dust. Heck, Tien is stated to be weaker than Krillin who also didn't train for a number of years. So this is hardly just Piccolo's problem. And again, this isn't Toei, that's Toriyama who decided that Piccolo wasn't going to train with Goku and Vegeta since he didn't do it in the Super manga either. In fact, I think Manga Piccolo said more or less the same thing as Anime Piccolo.
And if you're looking for an out of universe reason why Piccolo is not more important, it's the same reason why Gohan wasn't important until recently. He isn't important to the story and when he does become important he gets a boost. So far, Piccolo's role is the mentor figure to Gohan and him using skill to overcome raw power. That is his role until Toriyama decides to do something more with him. Even then, Piccolo got a huge boost this arc as we saw in Episode 88. He's only weak when compared to base form Goku who's ungodly strong even without transforming.
In-Universe explanation, the Namekians are strong, but are more limited compared to the Saiyans in terms of pure strength. It's the same reason why the humans are laughably weak despite if the story wanted to, they could power them up to.
We could also argue that he was desperate since he wanted Hit's siblings and more Saiyans.
The Hit line was obviously a joke and getting at least 1 more Saiyan was obvious. If he said he wanted Namekians it would ruin the surprise reveal in episode 102 which was suppose to be a 'What the heck' moment and was the point of the Zeno going through the teams on the tablet.
The scene was played as a joke, but Champa was serious. And said 'more' Saiyans, not just one more Saiyan. Ones that were stronger than Cabba specifically. Champa even wanted Frost back despite the BS he pulled at the last tournament, showing that he's indeed despite for anyone with strength. As for 'ruining' the surprise, the Namekians were revealed with no fanfare on a GodPad. The show didn't even play it as shock value, more like 'this is U6's team let's move on'. You want 'shock' value, look at Frost's introduction if you didn't read the spoilers before hand.