ckermit supports ssh these days (along with a lot of other things) and can handle file transfers over a connection using a variety of methods.
You don't really need support built into the ssh client itself. Having it built into any layer between the keyboard and the remote host will do.
KDE's Konsole had explicit ZMODEM support last time I used it.
screen (the terminal multiplexer) has support for executing a command and piping its inputs/outputs into the currently visible pty via the exec command (see the screen man page for details.) This works with X/Y/ZMODEM programs, but you have to start the screen session before you start SSH, and you may have to start ssh as
ssh -e none user@host in order to not have the escape character mess things up.
You might additionally want to look into ssh connection sharing. It's relatively easy to set up and allows use of multiple ssh, sftp, and scp sessions over a single connection so you don't have to reauthenticate. Doesn't help if you've daisy-chained through several hosts and suddenly need to transfer a file though.
In my experience, it's much more common to have base64 installed on random other machines than it is lrzsz these days. For small files it works quite well to base64 encode them and then "transfer" them via copy/paste. It technically works for large files too, but the transfer speed is rather low. It does have the advantage of not triggering escape characters for most connections though.