Not really. Gödel's theorem only proved that the *specific* kind of "mechanizati...

Not really. Gödel's theorem only proved that the specific kind of "mechanization of mathematics" that's implied by Hilbert's program as originally devised is impossible. The wiki has a nice section on varieties of Hilbert's program after Gödel https://en.wikipedia.org/wiki/Hilbert%27s_program#Hilbert%27... , citing (Zach, 2006) https://arxiv.org/abs/math/0508572

Of course from a slightly different POV there are plenty of practical implications, since e.g. unsolvability of the Halting problem can be viewed as a close variant of Gödel's results.