So only 2 of S codes, S.03 & S.04 which are ignition on and burner on, a bit strange that no other S code up when the burner stops, normally if the burner cuts out on say the boiler flowtemp exceeding its SP by 5C (where the boiler min output is greater than the heating demand) then you will get a S.08, the anticycle time.
You might look for a F (fault) code when next it stops, one would think though that if a fault code is flagged that you would have to reset the boiler?.
Not sure if available at your boiler output but you might try setting d.0, Heating Part Load, to say 10 or 12kw and see if any difference.
Where do you get these settings? "pump speed again today to PPA curve"