Kitty normally identifies itself as 'xterm-kitty' (in the environment variable TERM). The problem is that most terminal programs don't know that terminal type and use some less featured fallback. Most modern terminal implementations use a workaround it by pretending to be 'xterm-256color', with varying degrees of actual compatibility. (That's also what changing term in kitty.conf does).

Kitty's author really wants you to use the traditional solution of telling (programs on) the remote host what your terminal is capable of and how to use the capabilities by installing a new terminfo entry in ~/.terminfo/x/xterm-kitty on the remote side, which is what `kitty +kitten ssh [remotehost]` does. (It needs to be done only once, unless the file gets wiped on the remote end).