Well it would help if the GAs made an announcement every time and explained that anyone approaching the card reader before their Group is called will be denied boarding until their Group is called. If at all possible restate it in Spanish (many GAs at MIA, DFW, LAX, and NYC are bi-lingual). Then enforce the boarding. It won't stop all but it will stop some.
I'd also say that for those in Group 8 and 9 like Spirit give them an option-either pay $20 to check their large bag or $40 to carry it on. With no need for overhead space there'd be less boarding gate lice.