Paint the steel staircase yourself or leave it to the painter?

Water-based paint on a steel staircase? Was it primed by the manufacturer?

I have to smile a bit at the answers.

because as a layman it is quite time-consuming and not so easy?

We painted our steel stringer staircase ourselves: it was easier than a wall/a room and went quite quickly. In our case, even the provisional steps could stay on - why not? - after all, they cover what will also be covered later.
Am I right in assuming it is primed (reddish-brown matte)? Then just swing a round brush and apply the paint to your own taste.
Mask off the brackets beforehand, I think that’s the most complicated part ;)

Please do yourself a favor and use a paint roller instead of a round brush. Then I'll smirk back

Of course, you determine the effort and the result yourself. We had it done. I don't want to complain now because I wasn't in the mood for the work and I think we didn't pay more than 200€ for it. But if I had done it myself, the result would have been better.

Am I correct in assuming that it is primed (reddish-brown matte)?

Yes, that is exactly how it was installed with us. However, it is supposedly not a "real" primer. Of course, I am not really familiar with it.

we had sanded that a little (Alibi) and painted it twice over

Water-based paint on a steel staircase? Was it primed by the manufacturer?

Yes, exactly, it was primed with a reddish primer.