I know I'm supposed to say something like "we fight about it until one side gets mad and the other side puts their foot down." But what if it had gone the way you describe? I would have probably offered to retool a dragonborn or some other reptilian character race that's already in the campaign, and describe them as having descended from dragon turtles.
And since this is a game of what-ifs, the next question might be "What if I failed to convince the player to play something else?" It would then be my turn to compromise: I'd let the player play a tortle, but I wouldn't add the whole species to the campaign. I'd write the character as being from a small island far off the map, and the player's tortle would be the only one of his kind in the story. Interesting origin story about a shipwreck maybe, or a teleportation accident. NPCs reacting to him with all of the fascination and distrust that one would expect.
The next round of what-ifs might be "What if the player wants a huge, tortle-centric backstory? an extended tortle family and tortle-controlled territories? a tortle-based pantheon?" (I realize this is getting unhinged, but that's how what-ifs go.) In this increasingly-unlikely yarn, I'd try to find out why tortles--and tortles alone!--were so important to the player. I'd ask why turtle-people were needed so badly that I'm being asked to rework the entire campaign setting for them. And I'd endeavor to keep an open mind because who knows? Maybe they're right, and I've been missing out on the vast and colorful contributions that turtle-people could bring to our table.
There are more what-ifs, I'm sure. But what happened was, the player was only interested in tortle mechanics and so we found a way to make it happen to the satisfaction of all. That's the important part. Any what-ifs that come up are just steps in that process.
Obviously, other people have had different experiences. But I can't really speak for anyone else.
Nobody seemed to mind. The player was still playing a Tortle, as-written by Wizards of the Coast, in every way except the name and descriptions. It wasn't a "tortle," it was a "lake elf." He didn't "retreat into his shell," he "went into a defensive stance." Nothing was added or removed, and none of the math or mechanics changed. He still couldn't benefit from armor, still had disadvantage on Dex saves when in his "defensive stance," all that.