Glued stair steps are coming loose

Doesn't it look more like oak?

Yes, of course, not beech, but oak oiled twice.

So I have removed the step. As it looks, there are two problems here. On the one hand, as mentioned, the oil surface has come off, and on the other hand, in my opinion, the step does not fully rest on the adhesive. So, sand it down a bit and apply a bit more adhesive, right?

Yes of course, not beech, but oak oiled twice.

Either way - from a craft point of view, this is not a permanent connection. You have too many materials with too different properties and multilayer systems. The adhesion could be somewhat better if the oil layer is removed - which is rather not so easy with oil. You would have to remove the oil from the wood with solvents or mechanically. But even then you still have the problem that you have a material sandwich - wood works depending on room conditions and contains acid - adhesive moves a bit and can react with the acid in the wood - metal with coating does not move at all. If you rigidly connect the steps with screws to the metal, cracks can occur. There are reasons why traditional stairbuilders design the connections with clearance for flexibility.

What you see on the adhesive are the short wood fibers that it simply tore out. The adhesive apparently penetrated the oil well enough to reach the wood. However, through drying and shrinking, it simply ripped out the fibers. I would say this cannot hold. A better bond would be achieved if the wood were painted, but this cannot be reasonably done unilaterally. You have to imagine that the wooden step can swell or shrink by up to about 6% in width. These are enormous forces pulling on the adhesive bond. And you cannot prevent that – with painted wood you only reduce the process somewhat.

Or hammer elastic bushings into the wood and [die auf Stifte im Metall (gelötet?)]?

Or drive elastic bushings into the wood and put them on pins in the metal (soldered?)?

yes, you can try such solutions. But ultimately it is fiddling with a lot of effort to fulfill a design.