You’re allowed to carry bags onto the plane – but only if there’s enough space on board. That’s the main reason it’s better to board planes early, rather than being last onto the plane. Priority boarding is a benefit to ensure you get access to carry on bin space – even though no one wants to spend more time on board the aircraft than they have to.
Overhead bins are stuffed full, and passengers are forced to gate check their carry on bags, even though airlines are installing bigger overhead bins – bins that are supposed to have enough space for everyone. And it didn’t used to be this way!