I have been in contact with Shelly Support, and the Wall Display performance is now improved such that it is workable for me at the moment.
It appeared that the main cause of the poor performance was that the dedicated Home Assistant dashboard was too complicated, with too many elements and too complex graphical options.
I reduced the number of shown cards, and I removed all the historical graphs from the Sensor Cards such that only the current values are shown.
The response is now such that it is workable: when I now push a button on the touch screen the graphics of this button changes within 1 to 2 seconds.
And the number of crashes of the Shelly Stargate app is also reduced: it is now crashing 2 to 5 times a day instead of 30 times a day.
The issue with the direct Bluetooth connection between the Wall Display and the BLU H&T sensor seems to be more stable as well now: it now disconnects like once in a few days.
Shelly Support told me that the Home Assistance access via the Stargate app currently is still experimental, and they are actively working on improving the Stargate app, so hopefully the performance will improve further in the near future.