I'd translate "robotnik" as a "worker". A "person who is doing something" sounds a bit too general. Nevertheless, it's easy to guess that the root of the word is the same verb.
No, in Czech the meaning really is more specific and forced labor is not an overly bad translation, it's work done by serfs in medieval times.
It can be used to describe any work as heavy, but that's either in joke, or when used by people geographically close to Poland (typically referring to mining).
I vaguely remember that in English, we have words like “cattle,” French etymology, and “cow” Germanic, and the speculation is that it is because the aristocracy were French for much of England’s history (so, the French word is used to refer to cows as a sort of abstract resource to be considered in bulk).
I believe it's a similar thing.
Semi-educated guess based on historical facts:
Because of various (not least religious) reasons, Czech-speaking intelligentsia pretty much ceased to exist mid-17th century (fact, replaced by German/Latin) and only actual serfs spoke Czech, work and robota became defacto synonym (speculation). And when it became fashionable and cool to speak Czech 100-200 years later for the city-dwellers (fact), they probably felt the need to differentiate whatever they were doing as a job from the definitely uncool farmers of the countryside (speculation).
The evolution is the other way around - the original meaning of that root in proto-Slavic was obligatory work, and that one in turn is a derivative of a PIE word with the same meaning.
"Robotnik" means *manual* worker. E.g. someone fixing roads, or working in a factory.
"Pracownik" is a generic word for "employee". Can be an office worker or an manual worker.
"Robić" basically means "to do". While "pracować" means "to work". As you can notice one is more formal than another.
"Robota" is a less formal word, something closer to a "gig" (however it also means "work" or "job"). You will probably not find it in written texts. In writing "praca" (work) sounds more formal.
As a bonus, if you are a (manual) worker who complains and makes some not subtle digs about your job, you can say that you "have to go to your kołchoz tomorrow" ( https://en.wikipedia.org/wiki/Kolkhoz ). As the communist slogan said "work is your second home" after all :)
On a side note, do Czechs really like the polish word "pomidorek"? (Little tomato)