...because this is a much more delicate issue to dodge.
I am confident that 0 times infinity is anything, and this isn't even something new. The things I'm telling you your math teachers won't usually is that (1) infinity can be used just like any normal number (as long as you are aware of where the "general rules" come from), and (2) |1/0| = infinity.
Let's take (2) and apply it to the multiplication situation (zero times infinity):
0 * 1/0 = 0/0
Ask any math teacher what 0/0 is, and they will tell you "indeterminate". The only thing I'm pointing out is the (probable) historical meaning of the term "indeterminate". What it means is that the expression litterally cannot be determined. Here's what I mean:
0/0 = x
so by the definition of division:
0 = x * 0
x = anything
x cannot be determined/is indeterminate. The issue you raise can be resolved with an understanding that mutliplying zero times infinity is not adding zeros for a very long time, it is adding an infinite number of them and ending with a finite value, and as hard as it may be for you to accept, when you add an infinite number of zeros in a finite amount of time, you end up with an indeterminate answer. The only way you can side-step this is by declaring that "0 * 1/0" and "0/0" are not equivalent expressions, but such a declaration would go against the associative property, which you yourself brought up in that post.
Well, there is one other solution you could come to and that is that 0/0 = 0, but as I have shown, that contradicts the definition of zero, and if you did decide to accept this viewpoint, you wouldn't be preserving the modern viewpoint of highschool math teachers, which seems like it's something you'd rather do.. So take your pick:
1) 0 * 1/0 doesn't follow the associative property *just because*
2) 0/0 doesn't follow the very definition of division *just because*
3) 0 * 1/0 = 0/0 = indeterminate/anything