Trying to read into this, I'm getting the impression that all this was drastically dependent on time and place, and whether the institutions were able to afford to take in members without any money.
I'm most familiar with the early modern (not medieval) western european situations, where I think it pretty safe to say that most monks and nuns came from families with enough money to get them admitted. The matter was fought over a lot through the centuries, with the practice being considered simony at various points, but by the 1600s it was the norm for monasteries to not only accept but require a dowry for nuns, which could well be hundreds of pounds* for choir sisters. (All I am getting on lay sisters is that it was "less" and that they would be recruited when the monastery needed more hands on deck.) There were those like the Poor Clares who survived on donations exclusively and so didn't take dowry, but while this lowers the bar for access it raises the bar for how much support from those with spare cash is needed for the order to literally survive, so I'm doubtful those were popular among women who didn't assume that that support would always be found.
(*in pounds because the example I found was french/low countries institutions accepting english nuns)
Dowries weren't the only source of income - and this is what they might want more lay sisters for - because they also provided paid services and handiwork. They farmed, they span, they wove, they (increasingly...) educated the sons and daughters of the wealthy. They also had financial investments (especially in the low countries, go figure) and some had regular wealthy donors. The land they owned seems to have declined in financial importance, as the economies of the countries changed, they sold off more and more and there wasn't more land coming in to replace it.
This isn't to say they all operated totally financially rationally! Plenty of the records are from orders in financial crisis, and ideology was constantly in tension with practicality, but that wasn't a tension that was sustainable in a post-reformation, mercantilist Europe.
(A note on the other side of this financial equation: a peasant family being willing to give up a son or daughter to the church at all shows that they are doing pretty well by the standards of their class. Losing a working hand, even of the sort of labour a child can do, was not something many of them could afford. Still, I don't think peasant families who could stand to lose a son or daughter are exactly the target of this post...)
The impression I get is that it narrowed down over time. In the middle ages, yes, I think there wouldn't be enough people to populate the monastic orders if it were all the extra sons and daughters of nobility (though I would also love to see sources!! wish I could've found some admission records or something). The monasteries were financially solvent without the individuals they permitted being from wealthy families due to rents, taxes, work done by the lay brothers and sisters, and donations. This put food in the refectory and provided habits for most monks and nuns, and supported the expensive scholastic pursuits and sometimes far more comfortable lifestyles for a few, who were also those who wrote most of the historical record on the matter. Larping poverty isn't totally inaccurate describing those few, but they are heavily overrepresented in the sources. Most monks and especially nuns were not writing theological treatises or attending church councils, and rising in the ranks of the church was pretty much always tied to your societal position outside of it. (This class disparity within the church between normal monks/nuns and a privileged few is extremely well documented throughout various orders and times and places; the first source that comes to mind is account books showing what was eaten at the abbots table vs everyone else)
But, when and where that financial model broke down, the lay brothers and sisters who did not come with donations and connections were winnowed away. It isn't just that the writings and physical buildings of the most well-off monks and nuns are the ones that survived, the very structures of their orders evolved under economic pressure until only they remained.
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