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I agree on the second tank. It may also have been used for water hammer, or for hot water expansion.

On the well, though... not so sure. What I read is that eventually the well pump can boost the pressure to your set 50 psi, but it does it very slowly. Further, that when it does finally get there the pressure switch opens as it is supposed to.

This tells me that your surmise is quite correct: there is something going on which keeps your well and pump from keeping up, or even being close to keeping up.

So... what kind of pump are we dealing with here, since the possible problems are different for different arrangements. Is this a shallow well jet pump (the pump is in your basement, most likely, and there is a single suction pipe from the pump going down the well with a foot valve)? Is it a deep well jet pump (the pump is, again, in your basement, but there are two pipes down the well, and the jet itself is at the bottom or near the bottom of the well)? Or is it a submersible pump (the pump itself is down in the well somewhere)?

There are two problems which are more or less independent of the type of pump: first, the pump motor may be failing. This is rare, but does happen (a good submersible, for instance, has an expected life of more than 50 years). Related, the pump impellor or impellors may be damaged, usually from sand or silt in the well water. The second is far more serious: something may have changed in the well itself, or in the vicinity of the well, so that either standing water (when the pump is off) or the rate at which water can reenter the well (called recharge) has dropped. Either one will mean that your pump has to lift the water from a greater depth than previously, and will result in reduced flow (in the extreme, of course, the water level will drop to the pump intake or the pump itself, for a submersible, and result in no flow at all to speak of)(if a submersible is run dry, it will fail very quickly, by the way, so you don't want to do that).

With jet pumps -- either deep well or shallow well -- there is a strainer at the bottom of the pipe in the well. This should never get clogged, but there is a chance it might if, somehow, something might have gotten dropped into the well. There is also a foot valve down there which could get clogged or hang up partly closed, though again that is rare. Either could be checked, however, by pulling the drop pipe(s) and just taking a look.

In a deep well jet pump, it is possible that there is a leak in the motive pipe which takes water from the pump and runs it back down the well to the jet. That will reduce pumping efficiency a lot, but shouldn't lead to overheating of the pump motor itself (which you mentioned was a problem).

Submersible pumps have -- or should have -- a check valve at the pump. It is almost beyond rare for that to fail partly closed (although it isn't all that uncommon for it to fail open). Checking it, however, is not for the faint of heart, as it means pulling the entire drop pipe string and the pump.

If there is a problem with the well itself... has anything local happened which might have affected the groundwater? Construction in the vicinity? New foundation drains near the well? Yard drains? In any event, a competent well driller should be able to determine the static and operating water levels over time in the well, along with yield. That investigation should, however, not be undertaken until possible problems with the pump have been investigated.