Except that, like NaN (Not a Number), INF propagates meaningfully. Once you get an INF all future calculations using that number can only result in INF or NaN (or possibly 0 after dividing a normal number by INF, depending on the implementation. But that's probably okay).
MAX_INT/MIN_INT do not share that property, they're perfectly normal numbers at the limits of representation. The only sign you get of an error is that you have inconsistent data. If you have an INF or NaN showing up in your results it's pretty obvious that there's an error somewhere. MAX_INT though - do a little math on it and it will no longer be obvious that there was ever an error, especially if your valid data may be approaching the limits of your integer range.
Some specific failure cases:
+INF / +INF = NaN ----- MAX_INT / MAX_INT = 1 (You've just performed an undefined operation that completely removes any ability to make a reasonable approximation, where's the warning?)
+INF * 0 = NaN ----- MAX_INT*0 = 0 (same problem, you've done something that can't even be approximated, and it just disappears?!?)
0.0 / 0.0 = NaN ----- 0 / 0 = ??? (again, a completely undefined operation, what should you do?)
+INF / 7 = +INF ----- MAX_INT / 7 = some perfectly normal number (really!?!)
+INF + 1 = +INF ----- MAX_INT + 1 = MIN_INT (getting into some pretty esoteric theoretical mathematics to justify that...)
I agree it would be nice if there were a way to represent +INF, -INF, and NaN in integers, but doing so would greatly complicate integer calculations, which are extremely straightforward in silicon (well, division is a *little* complicated, but not remotely as complicated as floating point addition). And without silicon-level support you'd need to add branching conditionals to EVERY integer calculation, completely killing performance, as well as disrupting lots of traditional integer exploits such as rollover, bitwise operations, and others that are widely deployed as dramatically performance-boosting hacks.