> those cases where the two infinities are definitely not equal to each other

It's been a while since I was doing this in a classroom, but I feel like those things you're thinking are nonuniform infinities could just as well be thought of as entities comprised of infinity and a (perhaps implied) coefficient. Divide out infinity to reveal the coefficient. (And the same for powers/logs, etc.)

In this model, infinities are indeed uniform (quite similar to a constant), though they are often augmented in any of infinitely many ways.