They need to have GAs with good PA systems say firmly and loudly, 1st class only. If it does not say first class on your boarding pass, do not approach the gate. You will not be allowed to board until your zone is called. Please clear the way for 1st class passengers. They should then repeat this with Sky Priority. A little public shaming would soon make this problem a lot better. I think a lot of this problem stems from the fact that Americans are, in general, uncomfortable with the idea of status and privilege and therefore don't respect even deserved or lawful privilege.
That, and society has just deteriorated. Bah, humbug. old man muttering...