Why wouldn't they use public key cryptography for that?

Your satellite doesn't want to be sending out lots of different signals - due to a limited power budget.

So you have to send out one (or maybe a couple) of signals protected by a key.

Yes, you can distribute that key individually to clients using public key cryptography over the same link (and many services like pay TV do exactly that).

But fundamentally any client who is able to decrypt the main stream can also share the key with someone evil who can use that info to jam the same stream.