Yes, 'increase' is being used as a verb. If there are more enrollments over time, then enrollments are increasing, and someone observing this would "see enrollments increase".
If you added a 'that', like so:
...while the ACA continues to see that enrollments increase, there are a lot...
It makes it sound like the ACA is causing enrollments to increase. Right now, I am seeing that this question is answered. Someone observing me answer the question would be seeing this question answered.
To convert to this other structure, you are correct that you would remove the 's' from the end of enrollment. But that's not all. With just that change, we have:
...while the ACA continues to see enrollment increase, there are a lot...
In this sentence, "enrollment" represents the total number of students enrolled, rather than "enrollments" representing individual instances of people enrolling. So, "increase" is still a verb describing what that number is doing.
To make the full switch, you would also need to add an 's' to the end of 'increase':
...while the ACA continues to see enrollment increases, there are a lot...
Now the ACA is is seeing "increases", and the type of increases they are seeing are "enrollment increases".
Other possible phrasings include:
...while the ACA continues to see an increase in enrollment, there are a lot...
...while the ACA continues to see that enrollment is increasing, there are a lot...
...while the ACA continues to see increased enrollment, there are a lot...
Native speaker here – American English (New England, Boston area).
I may be grasping at phantoms here, but I have a very faint impression that when one uses "would always", it implies "on those occasions on which that person would do this thing, they always elected to do it in the following way", while, in contrast, "was always" implies that the person (or thing) so described was multiplying occasions to do the thing.
That is, using your examples,
She would always send me strange birthday gifts.
Suggests to me that on the occasion of your birthday, she could be relied upon to choose gifts that were strange, while
She was always sending me strange birthday gifts.
would suggest that she wasn't waiting for your birthday to send those strange dirthday gifts!
Likewise,
Sam and Mary would always choose the most exotic vacation destinations.
suggests that Sam and Mary's vacation destination choices were consistently exotic, while
Sam and Mary were always choosing the most exotic vacation destinations.
suggests that Sam and Mary have a very, very busy vacation calendar.
This
Ned would always show up at our house without calling first.
suggests that it was Ned's custom to drop by without calling first, while
Ned was always showing up at our house without calling first.
suggests Ned is making a serious nuisance of himself by imposing on us so very frequently. (Note that in this example, the "was always" formulation has a much strong negative valence than the comparative "would always.)
The difference, I think, is that used this was "was always" is hyperbole, while "would always" is merely a generalization.
Your description of "was always" + verb as implying a negative is correct, but it doesn't explain how that works. It is using hyperbole, which is a figure of speech which conveys valence by exaggeration. When someone says of someone "he was always doing that thing!" it is not usually meant literally; it is meant as exaggeration for effect.
In contrast, "would always" may well be meant literally, or close to literally. When somebody says "he would always do that thing!" they may well mean "whenever he did a thing, it would be that thing!"
By way of illustration: I had a friend who would always be late to parties. Famously so. Like, numerous times he arrived at parties when the last of the rest of the guests were saying good-bye to the hosts. Once, story has it, he showed up the day after the party. Yet I wouldn't say of him that he was always being late to parties because he didn't go to parties all that often. It wasn't like his life was full of party-going, only late. But, by gum, when he went to parties, he would be late; you could bank on it. So my saying of him that he would always be late to parties was perfectly, incontestably, literally true.
Best Answer
Yes, it's so (although, as I mentioned in my comment, those aren't even the only two ways the verb let can be used).
Cambridge labels these two uses of the word as to CAUSE and to ALLOW.
As for your second question:
it's important to remember that some uses of a word fall in between what a dictionary might define; one particular use of a word may not clearly map to one definition or the other. For example, the word strange can mean "unusual or unexpected," and it can mean "not familiar or well-known". But it's not hard to imagine a context where the word could be used to mean "unexpected and unfamiliar" – that wouldn't be so strange at all.
Given a choice between your two sentences, I think you'd want to use the word let:
That would sound rather natural in a conversation such as this one:
That's probably how I would word it. "Let him call me when he is back" doesn't sound quite as natural, but I'd still find it to be acceptable English.
As for:
that should probably be avoided. The "May he..." sounds like the start of a curse or a benediction:
I suspect you really mean:
But I still don't like that wording. That would be okay if you were disciplining a child:
Now, having said all of that, I want to address the crux of the matter. You're not really granting permission for someone to call you back, nor are you causing it. You're simply assuring the other person that "when he gets back" is an acceptable time to call. I think that's why I prefer can over let .
Notice how Cambridge defines the nuances of can: In addition to to be ABLE and to ALLOW – which are rather similar to the meanings of the verb let – there is also the meaning of to OFFER. I suspect that's what you're trying to do, to offer the individual the option to call upon his return.
I know this is a really long answer; I hope it is not excessive but instead clears things up for you. Indeed, the words can, let, and may have many overlapping meanings, and it's easy to see how they could be quite confusing in a context like this. I think you've asked a great question, one that would be interesting to natives and non-natives alike.