Haha, very picturesque, the smoke-filled room.
I just want to point out that I find this discussion very enticing. I think it's great that these questions are raised, and by no means am I just saying: look, this is how the text book has it, so neh (because then it really depends on where the text book is from; e.g., I know the French defines the square root of x as x approaches 0 as 0; I also know that some schools define 1 / 0 as ∞ to help students with solving limits -- but this is more of a shorthand for convenience than -- and I digress). It really is getting (and probably has to be) philosophical, which is great fun.
Moving on, I'll first try to describe one consensus (?) of the smoke-filled room. But only because I hope that it will help in moving the later discussions.
Infinity is a concept. So trying to multiply 0 by infinity is a lot like making this multiplication literally with a programming language (of course, a lot like, but not exactly like). You could usually safely declare 0 as int, but what would you declare infinity as? The very nature of its inability to be accounted for prevents it from being declared as int.
Going a bit deeper. Why do we have trouble accounting for infinity? What sort of infinity is it? This makes a difference. One tangible example would be the calculation of the probability of near-Earth object hitting Earth causing the end of the world as we know it. We know that it will be a number of extreme smallness (i.e., the probability of one near-Earth object hitting Earth) multiplied by a number of extreme largeness (i.e., the number of near-Earth objects for years to come). And the result? -- we really do not know, until we know it for sure. Another tangible example would be the measurement of something extremely light with an electronic scale. Each of the extremely light item would read as having the mass of 0 on the scale. And there really isn't any way to know if it really has 0 mass or it is just very light. But as we duplicate this extremely light item, which is infinitely small from the electronic scale's point of view, i.e., it is unable to comprehend or process how small it is, to the infinite amount of items, then there might be a comprehensible result -- or not -- we do not know, until we know it.
I think this vaguely insinuates why in calculus the answer would be indeterminable, as it really depends on how this infinity (and why it is infinite) is approached.