The phrase "jede Menge" literally translated means "every amount" but it doesn't mean any amount but a large one. Why does the phrase mean (almost) the opposite of the literal translation?
What I will write it's not how linguistics works, but I also see your point of "why cannot jede Menge equally mean arbitrarily small quantities"?
For what is worth: In some hyperbolic (ling.) figure jede Menge is some kind of unboundedness: given some bound L, then jede Menge X means that for some criterion you are using to measure X, measuring X exceeds L.
So far you would have symmetry (X's measure can be a lot smaller and OR lot larger than L), but Menge cannot shrink beyond nothing (smaller to the empty set you have nothing "emptier") as much as it can grow to infinity and this breaks the symmetry. Thus Menge means essentially large amounts.
For small amounts, you could use jede noch so kleine...
It's possibly a generalization from the literal meaning any amount, indeed.
It works for example to say that the big man will pay any amount, which implies practically, for all intents and purposes, a hefty sum.
A more convoluted derivation requires a fossilized phrase with independent development of the individual components outside the phrase. In particular, the je- is cognate with ewig and Engl. ever, so origin from intensifier usage is theoretically imaginable, but I am not aware of such a meaning with the determiner jede. It would in this sense be similar to so many ~ so manche, which also shows a determiner (so was formerly in the article paradigm with articles, thus Sie~ die for example) and diverging meaning (so manche meaning "some" in practice).
By the way, in the obverse direction we also see einiges (cp. any) used to mean "much", too.
PS: I think it's also possible that the -d- was parasitic, intrusive because of words like jederman, jederzeit, etc. We also have jemanden (to somebody), thus jemand (somebody), and we certainly have what sounds to me like je'e-Menge in Mundart. Thus it's probably derived from any of the severly erroded senses of je, ja, ganz, gar (Mundart janz, jar)¹, etc. without -d-, which must then have become inserted later on. The augment is similar to ever more, where the e- is thought to be akin to je-, but the details are less than clear. A relation to Frankish, later French (beaocup de "a lot of") is also imaginable, cp. En. all too many, Ger. allzuviele.
1: Der hat ja 'ne janze Menge ... "he has a whole lotta ...". I'm thinkin it possible that whole, holy shit, *Heil ... (long live ...), heilen (to heal, make whole) etc. belong here as well, cp. aye with a connotation pertaining to law, (potentially religios law) and jeh pertaining to longevity. See also jedoch ~ although (see how to interpret the meaning of 'jedoch' in terms of 'je' and 'doch'? NB: @imgonnamakeanaccounteventually is me). The etymologies of whole, holy, heal, Ger. hold, Holz and all are after all uncertain beyond Germanic, and the mere fact that eg. allerdings exists as well may be coincidence, an etymological doublette. all is thought to come from a sense "whole" (cf. Martin Haslelmath, 1995. Diachronic sources of ‘all’ and ‘every’)