Note that Eliminate does not hang (apparently go into an unsolvable state) with addition of irrelevant equations right up to the point that the third irrelevant equation gets added. Any idea why this might be?
Eliminate is in effect treating other variables as ones to solve for, and using polynomial algebra to eliminate T from the set of “polynomialized” expressions. In this last case it is apparently hitting some serious complexity in trying to compute a Groebner basis for the task at hand.
Stated differently, the function is nowhere near as smart as you think. A possible future task would be to recognize the special case where the elimination variables only appear linearly with respect to one another.