Check that your phone is not killing the background process.
A lot of manufacturers love to kill background apps to make your battery run for longer.
If you haven’t already you should check how to disable the battery optimization for KDEConnect. Unfortunately each manufacturer has its own settings and ways to do it, so I can’t offer more specific help, if this is the problem.
EDIT: It seems a lot of people already had answered making this point, for some reason the comments had not loaded when I initially replied 🤷♂️ Hope you’ll solve this! Cheers!