| This is flat out false |
Not false. Your response is subliminally talking about electrons in fixed orbits, not loose electrons. But I am talking about what loose electrons can do.
| the electrons do not just sit on the nuclei |
For "hydrogen in general", the electrons are in fixed orbits, and that's why they can't approach the nucleus arbitrarily closely. For loose electrons, they can approach the nucleus, but they cannot stay there because of the Uncertainty Principle. But when there are lots and lots of loose electrons available, any electron that must leave the close vicinity of a nucleus can be (temporarily) replaced by another. And when two nuclei approach each other, the place of greatest electrical attraction, between both nuclei and a loose electron, is the place midway between the two nuclei.
Finally, density is an average thing. It is perfectly possible for two nuclei, among many approximately holding an average distance apart, to approach each other more closely than the rest. Just like there is a bell curve of molecular kinetic energies with respect to the temperature of something, there is also a bell curve of average distances between nuclei, with respect to density. For ordinary hydrogen, with its very low percentage of deuterium, the probability of fusion is simply two low, most of the time, for it to happen when two random nuclei approach each other. But two deuterons have a much greater probability of fusing ---are you aware that Jupiter is reported to have a slightly higher temperature than it "should" have, based on its age and distance from the Sun? Rare fusions of deuterons in its metallic-hydrogen layer could be happening, sufficient to explain that.