> Wouldn't this likely be using asymmetric encryption though?
It isn't, but even if it was, not in a way that makes this attack significantly more difficult.
The problem you have is that GPS receivers are receiving one or more bit streams and passively discerning their position from the timing of those bit streams.
In something like TLS (except 0RTT early_data) both peers begin by introducing randomness to each session. So this connection must be the new connection I just created, if you try to send me packets from an earlier session, the random parameters are all wrong and it's unintelligible. Verifying this is called "freshness".
In contrast a passive receiver can't check for freshness. It just has to believe whatever it is told, no choice.
So, as an adversary I can spoof your position by simply feeding you stale data. I probably shouldn't try to persuade you this is actually the middle of the Pacific ocean not the Black Sea, nor that it's actually January rather than June, but I can certainly fudge things a little bit without too much trouble.