Latching timed circuit control

OK... I've been looking at various circuits using 555 to build the timed portion of the circuit but I'm not sure I need this level of complexity. Here's what I'm trying to achieve.

I'm building a battery powered filtration and evac system for another unit. I would like to have the circuit operational between 4.5vdc and 6vdc. This circuit will be in a non powered state until the activation button is pressed. Once active, the circuit should remain powered for about 3-5 min at which time it will shut off. A second switch can activate and hold the circuit from counting down via a pressure activated switch. It will close at aprox 30 lbs of pressure. Once released, the circuit would then remain powered for about 3-5 min. A third n.o. switch can be pressed to shut the circuit down and remove power to conserve battery. Correct me if I'm wrong, but I was thinking I could do this completely analog using a capacitor with a controlled charge rate to act as my timer.

Any thoughts?

That it could be done fully analog, of course, it can be done. Whether it is practical or economic, that is another matter.

I consider myself an analog guy, but this is one of those instances where the simplest microcontroller would achive those same results in a more straightforward and economic way. With better tolerance, too.