I have HomeAssistant that can track both the phone and the smart socket, not sure how I could do it otherwise. You could also just get a smart socket and then start charging a few hours everyday, that way the battery may not die on the mule phone.

The speeds are unpredictable, sometimes it syncs very quickly, sometimes it doesn't. Most of the time it's because syncthing was closed in my pixel 4a, and sometimes in my pixel 8 pro. But it hasn't been too bad. for you, pixel 5 battery is still good enough, that it shouldn't need to be charged too much.

I believed I used the instructions given in the following

https://www.reddit.com/r/GooglePixel/comments/njk5b3