Yes, the atomic hydrogen is probably mostly left over from the Big Bang. [Edited to add: Not sure how much that is true and how much present-day atomic hydrogen is the result of recombination.] And, yes, ${\rm H}_{2}$ does get dissociated by high-energy photons -- and also by cosmic rays, which can penetrate dense, dusty clouds that block most of the high-energy photons.
The real issue, as I understand it, is that it’s actually very difficult to make ${\rm H}_{2}$ by colliding two H atoms in the gas phase. This is because when two atoms collide, the resulting (temporary) molecule almost always has too much energy to remain stable and will very rapidly dissociate back into the two atoms -- unless you can somehow get rid of the excess energy before it does so.
Now, you can do this if the reaction actually involved producing extra components (like a more typical molecular reaction -- e.g., $AB + C \rightarrow AC + B$), since then the excess energy can be put into the kinetic energy of the products (while still preserving momentum). But in a reaction like ${\rm H} + {\rm H} \rightarrow {\rm H}_{2}$, there's no second body to allow that.
You could also do this if the "extra component" is some third body that happens to hit the other two at the same time (without becoming part of the molecule), since it can rebound with excess kinetic energy. But while this can work in dense gasses (e.g., in a laboratory on Earth), it will happen much too rarely in interstellar or intergalactic space.
You can also do this if the temporary molecule can emit a photon carrying away the excess energy. However, such transitions are strongly forbidden, especially in the case of a symmetric molecule like ${\rm H}_{2}$, which means that on average they will take too long to occur.
Finally, you can get rid of the energy if the reaction actually takes place on a surface that acts as a catalyst by absorbing the excess energy. This is thought to be the primary way ${\rm H}_2$ is formed in the interstellar medium: two H atoms that are adsorbed onto a dust grain combine on the grain's surface.
I mean, if the loose protons and electrons shot out by stars can (usually) find each other to form atoms, which is why neutral atomic hydrogen is more common between the stars and planets than ionized hydrogen, why can't the lonely atoms find each other to form $H_{2}$?
In addition to the fundamental problem outlined above, it's also easier (as Alchemista pointed out) for protons and electrons to "find each other" because they are attracted by their opposite electrical charges, something which doesn't happen in the case of neutral atoms.
