With the solid boosters currently on the Shuttle, they still have a little fuel left over at the end of their burn phase.
I think the main reason for this is that they want to make sure that thrust from both boosters ends within a small fraction of a second; otherwise, the induced yaw would destroy the vehicle.
But they can't make a solid booster burn so precisely that it exhausts its fuel within a small fraction of a second of a nominal period. So, they equip the booster with a pyrotechnically-actived vent at the top.
When the solid boost phase is complete, they blow the vents on the boosters simultaneously, and this cuts off thrust immediately even though there's a small bit fuel remaining (varying slightly among the boosters).
Maybe solid boosters can be designed to be capablie of this early in the burn, for emergency aborts.
Restarts are not feasible in any solid booster design I know of, but present-day designers don't see this as a problem, so long as they add a liquid-fuel booster which gives a part of the total thrust, and which burns beyond the duration of the solid boosters.
I didn't know that about the pop-top on the SRBs. Never thought about it, but it sure would need a solution, wouldn't it!