That is a fantastic and highly amusing speech, thank you for linking it.
• Epsilon Sandwiches https://www2.math.upenn.edu/~wilf/website/MAASpeech
In fact everything by Wilf that I've read is lovely: the paper "Recounting the rationals" (https://www2.math.upenn.edu/~wilf/website/recounting.pdf with Neil Calkin), and the book generatingfunctionology https://www2.math.upenn.edu/~wilf/DownldGF.html
For your example case, it's simply the matter of seeing multiplication as repeated addition. So, 2x is x+x, and from this follows:
2(x+y) = (x+y)+(x+y) = x+y+x+y = x+x+y+y = 2x+2y
However, for the other case, when we convert the multiplication by 2 to an addition and back:
2(xy) = (xy)+(xy) = xy+xy = 2xy
Now we can show that this works for n instead of two:
n(x+y) = (x+y)+..[n times]..+(x+y) = x+..[n times]..+x + y+..[n times]..+y = nx+ny
And for multiplication:
n(xy) = xy+..[n times]..+xy = nxy
Indeed, now that I've learned from a few times teaching the course that it always comes up, I will address it. But, if I spend my time building up basic-algebra proficiency at this level, then I'm never going to get into the meat of proving things. Perhaps the answer is to view the very act of writing your argument as a proof—and I like that perspective (because it genuinely is!); but, if I were to break that down to its full details, then I would have to write 2(x + y) = (x + y) + (x + y) = x + (y + (x + y)) = x + (y + (y + x)) = x + ((y + y) + x) = x + (x + (y + y)) = (x + x) + (y + y) = 2x + 2y, and writing down proofs of that sort risks alienating students who are eager for a conceptual picture, and guarantees giving the wrong idea of what sort of activity proving is. (And it also risks confusing why later I will say that, for a silly example, π(x + y) = πx + πy "by the distributive law"—I can't think of multiplication by pi as repeated addition, so I must cite the distributive law; and then why didn't I just do that before?)
It's not that they haven't been taught this; it's that they haven't conceptualised it, so that 2(x + y) = 2x + 2y to them is just a meaningless rule, and there's no particular reason why it should be true but not 2(xy) = (2x)(2y), or 2^(x + y) = 2^x + 2^y, or whatever. Of course ideally one would only have students in the course who have conceptualised this, but that's not how it happens, and it's not fair for me to teach the course to the students I wish I had.
In the case of having no intuition about algebraic manipulation, you can suggest a geometric interpretation to connect to intuition that's more likely to be there. For 2xy != 2x2y, draw two xy rectangles and one 2x by 2y rectangle.
Now the students all see the problem. Now they just have to connect the geometric intuition back to the algebra. This helps motivate the algebraic rules and shows why they must be what they are. Just the idea that geometric intuition exists -- that you can solve problems by putting pictures together in your head -- this isn't something every incoming freshman already consciously knows they have as a technique always available to them.
(This is just a wordy re-telling of Polya's "Draw a figure", from How to Solve It; if you haven't read, drop everything and get a copy.)