The assertion in the center is true, but irrelevant.
Humans like necessity, we ascribe it to tons of things. Most of those things are not really necessary. We learn from the exceptions all the time. The first time we drop a helium balloon and it does not fall, one necessity is replaced by another. Were we right or wrong before we encountered the exception?
If things we know are necessary firmly both personally and at a cultural level were not just wrong, science would stop. There would be nothing more to know.
So in what way is our ability to create incorrect assignments of necessity constantly proof of anything?
Hume doesn't have 'a problem'. He is not wrong. Necessity is a logical construct that we do not ever actually observe in reality.
We can observe strong patterns and deduce from them very high likelihoods, but we cannot observe necessity on any finite timeline. The exception might just lie in the future somewhere.
Hume does not rule out the ontological possibility of necessity. Nor does he prescribe a methodology. He just makes the deduction, mathematically obvious if one accepts basic probability theory, that we cannot know more than we can know. Certainty requires faith, and faith can always be either well placed, or misplaced.