The reason I stated the premise like this is because in my experience that is generally how new people are brought into a group. Basically an established player meets or knows someone that wants to play, there is an opening in the campaign (like starting a new campaign or a great spot to enter a campaign), and then they invite them over.
But all of that is beside the point. I have seen this example happen, and I have also been the person brought in. I have never, in my 40 years of being a DM or player, not seen at least one limitation set by the DM. Sometimes it's about backstory, other times it's about not allowing spells or classes, other times it's about not allowing a species, and other times it is about not allowing a certain rule. I have seen the reverse true too, giving players extra additions to their backstory, extra spells, extra classes, extra species, etc.
But this example is just a common way people join a gaming group. The host and DM are different people. It is a simple yes or no. There is no judgement on my part. I just want to see what people think.
I like your framing overall, but I think it has an accidental problem where the conflict is in the context of everyone already having sat down and gotten ready to play, combined with a social tendency that would often, I'd argue, drive people towards "gosh, I'm a guest / I'm a host, I better soften my approach to this situation". Maybe that's the point in terms of your common example/actual experience, in which case, fair enough. I've been reading this thread for the last ~2 weeks and finally got invested enough to post to point out that I think people are vacillating between a problem that speaks to what is allowed for a game in our hobby, and what is socially and logistically expedient or "optimal".
However...
Question: Based on the situation above, should the DM allow for the tabaxi?
A simple yes or no. That is all that is needed. It is not a trap, nor is it a gotchya. It is just to see where people are in this thread.
...I think this needs a "there is no universally right reaction" option as an answer to the question. Piles of pages ago (probably multiple times), proponents of both opinions presented valid reasons for a DM to say "yes" or "no, sorry" and for the player to say "cool, thanks", "oh, ok", "but what if...?", or "sorry, I'm walking". There is no shared context for the hypothetical in your question, so any of those exchanges could be right --- I could say "
of course not" for reasons that are absent anyone else's understanding of the situation, but I can't even picture these people's lives well enough to come up with my own answer. I'd even argue there
isn't a way to provide that shared context, and thus no judicious way to define what two people ought to do, absent suggesting there are RPG cops going table to table to enforce the law or DMs are being scored on their setting-building vs. player satisfaction.
I guess that's my overall answer, though --- I can't say what these people should do, and there is no law to guide us, and in my opinion that's a good thing.