e-SIM cards are not more private than physical SIM cards. Both of them bind to your phone, and the carrier will now know your IMEI and IMSI. Both of these can be tied to your phone even after you remove the SIM card.

So if you have a burner phone, and you attach it to a SIM card you own elsewhere, that burner is now tied to that identity.

If you’re worried about tracking put your phone into airplane mode, at least for Android devices that’s pretty good at disengaging from the towers. Then you won’t be tracked by the cell companies, but you’re limited to Wi-Fi.

But let’s go crazy, let’s say you buy a burner phone, and you only put eSims on it you buy anonymously, or SIM cards you buy with cash, that will still give your identity away by geographic proximity to your house. If you have the phone on in places that are connected to you, there will be location history showing you frequent those places. So if you’re going to go to this level, you better not use cellular anywhere that’s associated with you.