Why do they leave out the "On" here?
Beacause it's what Poles do. And what drives us crazy in Pol-Eng course
We often skip he/she/it/they pronouns, when subject is clear.
It is not as common as with I/thou/we/you.
But in the sentence itself it isn't really implied that whoever produces the chairs is male.
I would say the same about leaving out the pronouns if the gender was already implied, which it isn't. Native German-speaker here btw.
You can usually tell from the context, e.g. in a dialogue
-Co on robi? (What does he do? / What is he doing?)
-Produkuje krzesła. (He produces/is producing chairs.)
It would be very strange to just say 'produkuje krzesła' out of the blue.
In that sense, it's done in English as well.
What is he doing?
The pronoun shouldn't be omitted when the subject isn't clear--my opinion