I may be completely wrong - it has happened before, according to my wife ...
But I don't think pressing weight has a significant effect on the moisture level in the cheese. And I don't think the amount of moisture has a significant effect on whether the cheese is crumbly. Very very dry cheese will be harder, more suitable for grating than for slicing ... and if the cheese has aged long enough, it may develop the crystals that give it a somewhat granular character. But a cheese can be very dry, and still have a smooth and relatively flexible character; meanwhile, a very moist cheese such as a Caerphilly is normally quite crumbly.
Parmesans make the case. They are the driest of the dry cheeses ... and yet they are not crumbly in the sense of a Caerphilly or even a cheddar. At least the ones I've made, I can slice a thin slice, or shave off a thin slice with a potato peeler - something that would not work at all on a Caerphilly. Yes, it is so dry that it is easy to break the thin slice into pieces ... but this is a very different effect than the crumbliness that comes with low pH.
And here's the kicker - parmesans take some of the least weight to press. The curds are very warm, and the pH at pressing is relatively high, so they are eager to knit together. In fact, you have to be careful to press very gently and ramp up very slowly, lest you trap whey in the curds as they knit. (That is a different sort of "moisture" problem - but note that it is made worse, not better, by pressing at higher weights!)
Certainly the length of time pressing, if the cheese has not yet been salted, will affect the pH, simply because it will allow the pH to continue to develop. I would contend that the amount of weight during the pressing does not affect the pH in the slightest - unless it traps whey, which will allow the cheese to get sour. Definitely it is important, with any cheese that is salted after pressing, to remove from the press at the right pH - usually around 5.3-5.4. Otherwise, you WILL get a more crumbly cheese.
By contrast, a cheese that is milled and salted before pressing can be pressed as long as you want. And here the problem is to persuade the curds to knit, because generally these cheeses are milled when pH has reached 5.3-5.4 - and at that pH, the curds don't particularly want to knit together.
Again, I may be complete wrong ... the above reflects what I have come to understand by reading (especially Caldwell) and by making about 85 cheeses, but that makes me still a rank amateur next to Sailor or others with far more expertise.