We all know that in Doctor Who, every so often the Doc "regenerates" into a different form and we get a new actor. In VOY, Robert Picardo faithfully & skillfully played the ship's Doc from beginning to end. No hate for that. Just as a purely hypothetical question, if Picardo had wanted to move on and quit VOY, could you have accepted another EMH Doctor - even several EMH Doctors over the course of the show?
Robert Picardo clearly made the EMH very enjoyable, and made it his own. If he had left and a new actor came in to do the EMH role, either the new actor has to play it the same (which would not end well), or treknobabble would be used to justify "personality" changes - again, this would not end well. I remember "Ship in a Bottle" where the EMH goes through treknobabble back in the Alpha Quadrant where Judson Scott is taking a stolen experimental ship back to the Tal Shiar, and during the interim Kim and Paris - instead of anybody with any actual programming language skills - try to create a substitute and fail somewhat miserably, in scenes that really didn't do much and they made it look as if anyone could create a properly functioning interactive hologram in a span of hours or days, even if imperfect. A few more days or hours and they might get something more usable but it took longer than that for the EMH's creator to get v1 out, and in come two non-developers who manage to whip up something "almost there" so quickly? That's remarkable... but those scenes were for comic relief in an episode that's already largely comic relief anyway.
In other words, no option would have ended well unless the replacement actor was so much better and be a comparable fit without people drumming up parallels, which would be easier to do in that situation than compared to McCoy vs Pulaski where there's even less any similarity...
Ditto for Dax. Audiences were so attuned to Farrell's incarnation they forgot there's a big slug inside her belly that is the real character that any character can be a host for. Erzi, and I'm thankful they didn't try using the word "Ergo" as a name, was okay. If a symbiant slug can go from body to body, this brings up the other issues of staying in the same body for too long, thus cementing audience expectations and instantly disliking any new incarnation right out of the gate just because they're not the same (or going back to "copycat", there's a reason Doctor Who was quick to make a show about contrasting personas).
And, of course, if the new incarnation fails to win viewers over as those first few new episodes have much to do. I saw nothing wrong with Erzi, or how Nicole DeBoer played the new incarnation... some storylines were iffy but that's hardly her fault any more than it was anyone else's in the cast of players acting their roles out!
With VOY's technobabble, it wouldn't be hard to conceive of a reason: "The Doctor's matrix has become unstable, and we need to reinitialize his program. Parts of the Doctor will remain, but even things like his appearance and personality subroutines might be affected. It might happen repeatedly over the course of years, in fact. The program was made only for emergencies, and this is the result of overuse way beyond its original design."
Throughout VOY, the Doctor changed his appearance several times for several different reasons, and it demonstrated that it was a simple process for him to do it.
That part's the easiest and you wrote it best. I even imagined Janeway saying it, and it has the flair of authenticity to it and could potentially gloss over the technicalities that I meandered through earlier with ease. The choice of actor is still the key factor to make such a replacement work but if it ain't broke don't fix it? (Meaning, fixing it tends to make big messes in the end...)
In DS9, Dax presented a special set of situations that involved previous lovers and all kinds of nuances that got explored to varying degrees. A similar concept could've been used with this - if the circumstances had required it. Again, Picardo did a fine job, this is just a hypothetical for fun.
Piggybacking on thread "The Worf Effect," who would you have put in the Doctor's role if a "regeneration" were required? Not just your favorite Trek character, but what species or other categories would you draw from for additional interesting stories from Sickbay?
P.S. This show had a whole lot of Roberts on it!
Any actor I'd want to see? Typecasting concerns aside, I'd peg for Robin Williams, partly because it's not easy to typecast him but we still know he has a genuine sharp wit that's pliable and adaptable to many situations as well as being serious. But he was too much into movies and there are reasons movie actors don't go back to TV, which goes right back to typecasting...
For categories, maybe the other obvious low-hanging fruit on the tree: James Bond. Just swap actors (and background scenarios such as the cold war, relic of the ended cold war but is now just another spy, or terrorist organization (revealed to be SPECTRE) whenever necessary. Or even favored games, since the franchised got Americanized by replacing his game of Baccarat with... Poker?! (Amazingly, they made it work but it's a nitpick nonetheless since it Craig would have made it work with Baccarat anyway, there was no reason to change it. Why not go all the way, change it to "Go Fish" instead so even more in the audience will understand without having to look it up or have dialogue smartly conveying gameplay without dumbing down the audience in the process, but I digress.) James Bond never had any explanation, but it's impossible to say he's the same bloke who was married and had his wife killed, each new actor put in, in a way, becomes its own parallel universe version, with past events alluded to but in no direct way. "Licence to Kill" did mention his marriage and wife killed, as had "For Your Eyes Only", but neither dared attempt Bond doing a facelift or anything else dumb to explain it in-universe. Bond always did it right for its 6 theatrical actors playing him. James Bond is a code name that nobody knows (unless you're in "Diamonds are Forever" where they forget that fact and now everybody knows him and treats him as a pop star) but James Bond is a subcategory within the 00-Agent group, of which 9 such Agents exist (or 10 if we use a 0-based numbering scheme but nobody wants to be Agent Double-O Zero... unless you're in the 1995 continuation of "Get Smart", also featuring EMHv2 himself, Andy Dick and he's actually funny in that show, if you didn't think he was funny in VOY.) But I digress. the 00 group allows a lot more to go on in the top category that contains the subcategories and female 00-Agents have existed since 1965 (a photo op showing who has been pegged as 004 was female, so I fail to understand what the furore is about this issue... but I digress again.)