I'll go for the contrary answer: this may actually be a (stupid) miscommunication and not actually be a red flag.
Don't get me wrong: if they're asking you to quit your current job and only then get an offer - yeah, that's a terrible idea.
But from your question, they're insisting on simply knowing your joining date - or when you'd be able to start. Giving this answer does not require you to quit your current job, or anything of the sort. In your shoes, I'd simply reply something like:
"Well, if you can get the offer letter to me by the end of the week
and I find it acceptable, I think I'd be able to start March 1st -
that give me time afterwards to give notice and get everything in
order from my side."
Aka, you're not quitting until you accept an offer; you're not pressured into accepting a lowball offer or a different role than you expect. And if all of this is a simple communications issue, it'll result with them sending an acceptable job offer/contract. (If it isn't miscommunication, and they're being underhanded - well, that'll be clear pretty quickly and you won't be an awkward position of having to accept a substandard offering from them.)