Nope, not "more like". Strictly speaking, that's less accurate and possibly idiomatic. "tud" with infinitives simply means "can" or "be able" and that's that, I don't know why people can't accept that or get this misconception that this use must be explained using the "know" meaning...
Your sentence would be: "Nem tudják, hogy kell festeni".
You're getting confused by two different English senses of the phrase "can't paint".
In English, "I/they can't paint" is more naturally understood to mean "I/they don't know how to paint", so it's a better translation of "nem tudok/nak festeni". You are correct that in certain contexts, "can't" also means not permitted, but generally that context would be specified - "you can't paint here" or (speaking to a small child) "you can't paint until you've finished your breakfast". In these contexts, yes, festhetnek would be better (at least to my limited understanding).
Now that's why I'm particularly angry with people spreading this misconception of "tud csinálni" = "know how to do"...
Nope, it's not more correct. "Tud csinálni" is the closest to "be able to do" and vice versa - "be able to do" is the closest to "tud csinálni". If you wanted to say "know how to do" in particular, that would straight up be "tudja, hogy kell (csinálni)".
Having said that, "-hat"/"-het" is about theoretical possibility, maybe closer to "may". "I may paint" both as in "I'm allowed to paint" and in "There is a chance I (will) paint". Not exactly the same as it's less common for marking probability but that's the closest I can get with one word.