"There is no privacy concern, really..." Except for the network operator, who needs to track a SIM card not a phone, but who can track you across networks and SIM cards if he has the IMEI. There is no reason the IMEI needs to be stable.

The network operator does NOT need to know who you are, even if you live in a repressive country that mandates tying ID to mobile phone lines. Get a SIM card in person and top up in cash, or use a virtual credit card, or pay in cryptocurrency for an eSIM, or get a subscription in a less oppressive country and roam.

Invisv is a great suggestion.