ᐅ Which type of hardwood flooring would you recommend?

Paint is common. You don’t have to imagine a thick, glossy layer underneath. There are also matte paints, which in my opinion the average person can hardly tell apart from oil finishes.
Oiled surfaces tend to be a bit rougher to the touch.

These days, when people talk about parquet, they usually mean click-lock flooring with a few millimeters (inches) of real wood on the surface. It comes ready-made with a finish already applied. That’s minimal work for professionals, who would probably like to charge for a few extra days to apply the finish themselves...

We have solid wood, not click-lock or two-layer systems.

There is basically nothing wrong with using lacquer, even on solid wood. The parquet flooring in my parents’ house is made of solid wood with lacquer. It withstood four children exceptionally well. After the youngest daughter moved out, the floor was sanded down and then oiled – which turned out to be less durable.