Macos – Equivalent of ‘clear’ command when using tail -f

That's unusual. I'd start by seeing whether Ctrl-L works (ie, zsh knows your terminal and can issue a clear-screen) and running echo $TERM to see what terminal type zsh received from Terminal.app; I have Terminal.app send xterm-color and that seems to be portable and accepted.

Another thing to check is what happens when you type Ctrl-H, but I thought that all modern systems had settled on 0x7F for backspace.

[post-update with screen information:]

Bash has more built-in hacks for assuming that missing terminal types are ANSI-ish, but your root problem is that the $TERM is unrecognised. This will break most apps that want to issue control directives to the terminal emulator. Override $TERM in your ~/.zprofile:

case ${TERM} in
  screen-256color) TERM=screen.xterm-xfree86
  ;;
esac

This is part of the problem of termcap/terminfo -- it relies upon everyone having the same names, everyone being updated, etc. No negotiation is done.