TyroPicard
I think you have answered your own question. The result of selecting more flap would be reduced pitch authority which is why you did not do it on the types you have flown.
As to why there would be reduced pitch authority it is again as you say caused by the change of downwash from the bigger flap angle.
If there was any change of trim with the final stage of flap then part of your elevator capability would be absorbed in countering that trim change (which ever direction was involved) leaving less for you to use to manoeuvre the aircraft. If there was no trim change with the final stage of flap then I would suggest there would be no loss of pitch authority.
But I suspect you had sussed all that and were looking for confirmation.