I find myself in between. I've seen many terminfo's set to other things, which don't work, or have to be set every time I'm connecting to a new server which is too much extra work for every new server in my opinion. So I always just set it to xterm-256color and forget it
I use rxvt-unicode which sets $TERM to "rxvt-unicode-256color," which is not recognized by a number of server to which I have to connect to from time to time.
There's a whole story about that, part of which is that at the time terminfo had to cater for old systems where there's a limit on the length of a terminal type name. But the same applies as for a missing kitty terminfo entry. The terminfo database is designed so that one can copy records around, and one can put them in a directory in one's home directory if one does not have administrative access to update the system database.
Debian systems have had an rxvt-unicode-256color entry for approaching 10 years, now. FreeBSD for about the same.