I just cleared an Alchemist entirely through the 43b prerelease main campaign.
Golem couldn't learn Prodigies. Never got any points for them, ever. Couldn't use the Alchemist's points for them either.
I was, in fact, quite disappointed that he doesn't. I'd kinda like him to get one at 50, though maybe that's for a more long term overhaul of Alchemist, when you can nerf them without totally gutting them... But that's a side issue.
Golem always could learn Celestial/Light, to boot, by giving him as many Generic rewards as possible and giving him the category from an Escort, then using Wyrm Blood on him. And...he doesn't get Generic points through leveling, normally. He's sitting on 0 of them right now, on that cleared file, which is all I've ever heard anyone talking about him getting. Since obviously you could easily abuse Wyrm's Blood to get a large bonus in a Generic category, otherwise, so a fair deal of builds would do it.
Are you sure this isn't some ID-only anomaly?
Edit: As to the theoretical exercise of what a Golem qualifies for(Which, yes, are calculated separately... I kinda thought Golem did get Prodigies so I was paying attention to things like this.

), a Golem does not get 20 points in Stamina or Spells without dumping every single Class point they get, along with several Class boosting items, into one or the other category, from what I recall from mathing it out, in the main campaign. They just can't manage to get the 20 class points otherwise. It's probably not worth losing the excellent Eye Beams/Molten Skin or the ability to close the distance fast with the physical skills to get a specific Prodigy.
They can fulfill the stat and elemental reqs, but there's literally two Prodigies I consider actually good for them, that they're likely to get-Arcane Might and Irresistable Sun. At least, in the main campaign. Assuming they can learn them, which they couldn't for me, because they never got points.
