KDE Connect SMS has always been problematic for me, probably because I have a TON of messages.
Sometimes it works like a breeze, sometimes it doesn’t.
Sometimes deleting all of the kdeconnect data from ~/.config and ~/.cache helps, sometimes it doesn’t.
A lot of times, I end up just using the clipboard sync function to type out a message on my computer and paste it into the SMS app, or maybe use the remote keyboard function.
I found a merge request from a few months back that purported to fix the issue, but work on it seems to have stalled.
Interesting, I just tried killing both kdeconnect processes (kdeconnectd and kdeconnect-sms, then deleting both the kdeconnect.sms and kdeconnect.daemon folders under ~/.cache, and that seems to have brought it back to life in an instant.
No guarantees that it will work for you (or even for me a couple days from now), but it’s worth trying. 