It's literally not possible because the mobile network needs to know where your phone is so it can route packets to the tower that you're connected.
The whole problem is that a person's identity is more and more entangled with a tracking device.
Since then I've been very skeptical of all claims of what you can "just" do to avoid being tracked. This problem can't be solved by individual action or paranoia it's too big and complex now.
I'm not sure what the threat model here is. If the "government actively trying to build a case against them" involves government agents following you in unmarked vans, then the government being able to track you through your SIM should be the least of your worries. If you're merely on some government watchlist (ie. there aren't agents following you), I don't see how it would be hard to cleanly buy a burner phone/sim, especially in the post pandemic age where face masks are socially acceptable.
You can't really protect yourself from the US government in any meaningful sense if they decide to end or ruin your life, but what I described above seems like a reasonable thing that people would want to do for reasons both legal and not.
Pure conjecture on my part, and I haven’t worked in this area but I had lots of friends in gov security and mobile standards groups.
Nobody I’ve talked to thinks it’s possible to stay anonymous on any commercial cellular network.
You might be able to push a few anonymous phone calls or surf a few hours but soon your typical browsing habits, app usage and contacts will match you up with similar existing profiles based purely on the metadata.
Honestly just narrowing it down to a few dozen people is good enough to get unmasked.