The only actual creative change that corresponds to the Piccolo/Saiyan split in the manga is Toriyama's editor. His first editor, Kazuhiko Torishima, moved on after the 23rd Tenkaichi Budokai. His next editor, Yū Kondō, was there until Cell absorbed #18.
That first change in editor cleanly matches the DB/Z split in the anime, which, naturally, comes with a new opening theme song. The second change in editor comes pretty close to lining up with what Gaffer Tape aptly described as the next most logical split point: the Cell/Boo boundary, which, conveniently, also serves as another boundary between two different openings.
Dragon Ball, the comic book written and illustrated by Akira Toriyama, is one story. But if you felt the need to break it up into sections, I think the editorial changes largely show where to do that: into thirds, rather than halves. Deliberately or not, the anime pretty much does exactly this.
On the topic of tonal genre, though, I think it's more important to look at the story in one piece, rather than in three distinct sections. I see Dragon Ball's tone less as something that changes over time, and more as something that receives additional layers over time.
At its lowest, most fundamental, and foundational layer, Dragon Ball is a comedy. Akira Toriyama, by trade, writes funny comics. He's a comedian. His art is cartoony. Characters are named in accordance with elaborate and interrelated systems of name puns. His panel layout is rooted in comedic timing.
Naturally, being a fantasy martial arts series, it's also filled with action. This is the next layer up. The art gradually becomes more angular, largely in order to better facilitate action scenes, but this is still an angular deformation of his cartoony art style. Despite drawing fights and action sequences, the paneling is still firmly rooted in his comedic sensibilities. In a way, the fights are illustrated using, and flow with, the timing of jokes and their punchlines (no pun intended).
Finally, we have the drama. Things get genuinely grim for the first time with Tao Pai Pai, and hit the next major stops with Tenshinhan and Piccolo, before giving way to the likes of Vegeta, Freeza, and Cell. In other words, theres a very gradual increase in the density of the "drama layer", before we come out on the other end with the Boo arc. The final arc is arguably a huge return to form, with gags finally regaining their previously held prevalence, now sitting side by side with the drama. You could almost describe Dragon Ball as "a comedy, with a bell curve of drama in the middle".
It's worth noting that, even in this bell curve, the foundational layer of comedy never goes away: it's a dramatic story, but one that is presented by a comedian, in his cartoon-rooted art style, featuring fights that retain his gag timing, involving characters named after musical instruments, food items, food containers, and so on. This is to say nothing of characters like Kaio, the Ginyu, and Mr. Satan, who more directly inject comedy into arcs that otherwise (seemingly effortlessly) trick the reader into taking deathly seriously.
The far end of this curve, the Boo arc, also does a lot of self parody and bucks a lot of conventions that previous arcs had set up. That "bell curve of drama" almost acts like a setup for a joke, to which the Boo arc itself is the punchline. Meaning that even though Dragon Ball gets serious as fuck, and for quite a while, that seriousness acts, in the big picture, in service of the larger joke that the work as a whole represents. It builds, and builds, getting increasingly serious, stringing the reader along, tempting them to get more and more invested, getting them used to a certain narrative rhythm, only for the final arc to boldly declare: Gotcha!
I don't know if anyone here is familiar with Dave Chappelle, but he is a comedian. He's been in films, has had several stand up specials, and was most famously the namesake of Comedy Central's Chappelle's Show. He eventually left the show on very bad terms and made extensive efforts to avoid attention from the media and public. He recently had a slew of stand up specials on Netflix. At the very end of the last one, he tells a story about Iceberg Slim, a pimp who wrote an autobiography. There's a story in the book that Chappelle says encapsulates his experience leading up to his grand departure from the show. That bit feels incredibly similar to this reading of Dragon Ball that I'm presenting: it's a relatively very serious portion of an otherwise comedic performance, presented by a comedian, flexing his aptitude for comedic timing, and it still ultimately ends with laughs. I'd recommend the special itself for anyone who enjoys comedy.