Saeco XSmall no water over coffee spout

Moin,
I am currently trying to repair my Saeco Xsmall and am encountering the following problem:
Neither during coffee dispensing nor during the simple rinsing process does anything come out of the coffee spout, it remains completely dry. The following steps have been taken so far:
1. Pump check: pump is fine, as hot water output through the nozzle works fine.
2. brew group: I have completely cleaned the brew group and especially the strainer.

I think it is the hot water control valve. If it is not installed, the amount flowing through looks like this:
https://drive.google.com/file/d/1XcGsevFFZx44AxZ4eX46qquGTmyVerJ6/view?usp=sharing

However, if it is installed, hardly anything comes out, just very little drop by drop:
https://drive.google.com/file/d/1AzOA_2tCudZVtr0WV8KMY3ODtZhnfGWd/view?usp=sharing

Could this be the problem and is there any way to fix the venitl?
Or does it need to be replaced altogether?

My guess is more likely the pump. If you draw hot water, the pump does not have to work much. With the valve, the pump sounds exactly the same as without the valve.
Therefore, I think that the no longer brings proper pressure.