Note: this is my probably flawed and possibly wrong understanding, not objective fact
It’s a verb phrase (you can think of it as a set expression, but I don’t think that’s what’s at play here).
With regards to your examples, the reason for not using the て form is those adverbs modify うまく, not 行く. For example with 多くうまく行く, things do not “go frequently and go well” as 多くてうまく行く would imply, they go well often. Same with 間違いなくうまく行く - what’s unmistakable isn’t that things go, it’s that they go well.
Consider a different verb phrase with a different adverbs:
よくて速く走る - I run often, and I run fast
よく速く走る - I often run fast
Both are correct, but they mean different things, and in the case of うまく行く it’s just not often the case that a second adverb would also modify 行く as opposed to うまく, I suppose.
One of the cases I find is, for instance ちょろくてうまく行く, which is one of those few cases where it does make sense for both to act on いく- it goes both well and easily.