Craftsman demands "pitching in"

If he hadn't wanted the order, he could have refused it. It's still a free country where no one is forced into anything. The facts (that it is a staircase from another carpenter) were known beforehand and the craftsman voluntarily submitted the offer at exactly that amount. We had 3 other offers for the installation, all roughly around the same price. So this does not seem to be an unusually high wage. We chose this craftsman because we already knew him from the window installation – he worked reasonably there.

Construction expert, that sounds as if the "usual" portals were devil's work. We have already recruited several craftsmen from there and it has (almost) always worked out well. These were all companies from the local area, which we otherwise would have had to call around laboriously. Placing an ad – and nothing else is MyHammer etc – is simply more relaxed because the craftsmen who have time and are interested in the job get in touch. What speaks against it?

Why didn’t the carpenter who made the staircase install the staircase?

is simply more relaxed there because the craftsmen get in touch, have the time and are interested in the job.
What speaks against it?

Well ... interest...
I once submitted my offers there myself as a photographer. Fortunately, I’m not dependent on this side job, because at absolute dumping prices, which definitely ruin the market and the existence of freelancers imposed by some hobby snapshooters, the work is no longer fun. Respect, which is earned through price-performance, does not exist.
I received one job, MyHammer earned a share - but at that dumping price, I was not willing to work out small details as I usually do.

This will probably be reflected with other craftsmen as well – you can see it

And to the first question: yes, we were also asked to lend a hand sometimes. But only occasionally, since we were also on the construction site. In return, the craftsmen occasionally provided us with this or that tool for the EL or, for example, helped carry the carpet roller upstairs
We were definitely not planned as student assistants.

Why didn’t the carpenter who made the staircase also install it?

Because the carpenter has his workshop about 600 km north of us. He was the only one who submitted an offer for a staircase the way we wanted it. The other offers were either hopelessly overpriced (6000 euros for a secondary staircase is simply out of the question) or some kind of compromise we didn’t want to accept, e.g. not saddle-shaped or without risers or too narrow in the tread or clearly lacquered.

However, I haven’t had bad experiences with this so far. Also the heating (boiler) was sold to me by A and installed by B (however, B was recommended by A), likewise the air conditioning (bought at Metro, local climate company installed it, everything went great), or the front door (built by a carpenter in Bavaria, installed by carpenters from nearby). In each case, I looked for the installer before buying and only ordered afterwards.
It was the same with the staircase. I didn’t buy until I found someone who would install it. If I hadn’t found anyone for the staircase, it would probably have been a kit system staircase (center stringer).

If you always have to/want to watch every penny, things like this just happen. Of course, it is more or less audacious on the part of the carpenter fitting the installation to involve you. But you have done it now, and he wants to be paid for his work. If he has removed the staircase (also with your help), then he should also get his wage for it. And he installed the fall protection, so why shouldn't he pay for it? Of course, he must not charge for the materials.

..... why shouldn't he pay then?

"to get paid" was what I meant of course........ those who can write clearly have an advantage