I think that explanation is only about Hydrogen molecules and hydrogen atoms, not Hydrogen vs. Helium.

There's a free textbook available here [1] that explains that atomic sizes decrease across the periodic table and increase down. In short, it decreases across a horizontal period because more electrons are in the same orbital shell (really a cloud), increasing the negative nuclear charge of the cloud, which is more attractive to the positive atomic nucleus, so the whole cloud is squeezed together, making the whole atom smaller.

[1]: https://chem.libretexts.org/Bookshelves/General_Chemistry/Bo...

This seems unintuitive as I would expect one electron by itself to be more closely attracted to a positive nucleus instead of when there are two or more fighting to remain in "orbit".

Your definition would make sense to me if you can think of electrons adding to an over all single electrical charge instead of individual electrons, but in that case, I have to ask, what is the original reason a single electron is repulsed by the atom. I.E. why does it stay so far away until more electrons are added. What is the force keeping electrons from falling into the nucleus in the first place and have an "orbital" or "cloud" at its set distance (like one Angstrom or Bohr)?

My instinctive guess is that it has to do with the spins of two electrons "balancing" each other keeping them closer to the nucleus.

My understanding is that spin leads to a slightly different electrical force between up and down spinning electrons due to weak force symmetry breaking or singing to that effect. It's more like they have tiny electrical differences rather than canceling out any charge unless I'm mistaken.