How To Hold Power ON For Some Time Without Using A Microcontroller or 555?
Your goals are contradictory.


  my primary concerns are NOT using a microcontroller or 555,


Okay. Not a real engineering goal but maybe an interesting goal for learning from. You'll find lots of monostable circuits online if you just google image search for them. It's also still possible to design digital logic from single gate building blocks.


  holding the Output steady during the entire "unstable state"


You can probably solve this by using a buffer between the logic circuit's output and the load. The buffer might be a simple as a single MOSFET.


  using no or minimum quiescent current during the "stable state"


Almost certainly the best way to do this is using a microcontroller instead of a monostable. Many microcontrollers can enter a very low power sleep state, and then respond when they receive an interrupt (for example due to a pressure switch being activated).


  for a minute (give or take)


It's generally tricky to get a monostable to keep time accurately over such a long period. Again, the low power, low cost solution is likely a microcontroller. You could also design a digital circuit using discrete logic (start with a counter), but it will almost definitely consume more quiescent power than the microcontroller solution.

A general tip: When you're designing something for an application you've never tried before, first make a working design --- just one that has the basic required functionality --- then once you know how that performs start thinking about how to optimize it for speed or power or whatever's important in that application.