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.
And of course, even on WiFi 'Google retains a detailed map of known Wi-Fi networks and access points. By knowing the exact location of these networks, and your proximity to them, its location services can gauge your location with roughly 30 feet of accuracy.'
I do get there are privacy ramifications to this, but the alternative is having to wait like 2+ minutes for a accurate gps lock every time your phone needs location.
I do not mean this in a snarky way, everyone has their own priorities. I don't use location for anything. If I did, having to wait would be an inconvenience.
And of course, even on WiFi 'Google retains a detailed map of known Wi-Fi networks and access points. By knowing the exact location of these networks, and your proximity to them, its location services can gauge your location with roughly 30 feet of accuracy.'