Thanks @teezeh and @aspergerguy.
I just discovered another important clue:
When my phone occasionally detects and initiates my SIM card, slight mechanical agitations will make my SIM card briefly “disappear”. For example, if I hold my Fairphone 3 with one hand, and tap on it with my other hand, the associated item in my status bar would briefly turn to “no SIM card” for a few seconds before re-detecting the card and re-initiating a connection.
@teezeh: Is it still possible that a new SIM card would solve this problem? Or does this point towards something with the phone itself?