There is no general answer to OP's question. What carrier, what airport, and even what gate.

While the carrier really matters, the latter may affect the practicality of the policy's implementation.

AA, by way of example, if it has two lanes available, will always have one of the lanes for priority passengers. If you arrive after Zone 4, you may still enter the priority lane and the agent should call you forward by unhooking the tensor blocking it. If there are only a few people in the "general" lane, it seems not worth it. But, later in the process, when the general lane becomes full, your ability to use the priority lane means less time in line and seems quite worth it. That, of course does not work at gates which lack the space for two lanes.

Other carriers vary, but the general principle is that "priority" means "priority".