Solar thermal system efficiency improvement

Optimizing something at a St is like casting pearls before swine. In summer you have too much anyway, in winter basically nothing.

The optimization then probably consists in not additionally charging the storage in summer alongside the ST.

This is how it looks, because if you just let it run, both will keep heating the storage from time to time when it is empty, regardless of whether it makes sense or not. The district heating after the morning shower and the sun only a little later, but by then the money has already been spent. This is exactly what should be prevented here.

The ST should heat all the water as long as there is sun. Or rather, the district heating should only start when there is no longer any prospect of charging the storage from the ST.

That is exactly why the heating only charges the storage tank in the morning before getting up, because there is no sun yet.

Does the afternoon sun perhaps not suffice to last through the night, or is there an extensive shower in the evening and the storage is then empty?

Just out of curiosity...

More the latter... when the daughter is in bed, the wife goes into the tub. That's enough in the morning for brushing teeth but not for a proper shower.