Using its own TERM is a deliberate design decision. I don't remember how to fix the terminal database, but it's pretty easy (your favorite search engine or LLM should be able to help you there).
i had to do this for ssh
host * SetEnv TERM=xterm-256color
I definitely have had to poke at things a bit, even where they "should" work. I use salt in my homelab and eventually got the xterm-ghostty.terminfo file and I put it in my baseline salt config, then you run 'tic -x xterm-ghostty.terminfo'.
Or the manual, which describes the features for automatically handling it.