I suspect that the German speaking countries, when they built their systems, just copied what the Hungarians were already doing (no turnstiles).
Even our one metro line that does have tap and go single ride tickets, there is no turnstile, just a card reader that takes a few seconds to use. To be honest, I think its on that one metro because its the one the tourist use and its for their convenience; not for Hungarians (they use multiple day passes for the most part).
I can’t find a written explanation of why there are not turnstiles in Budapest, but as the metro trains run every 3 to 5 minutes I doubt it has much to do with breaking a stride. As Hungarians don't have a 3 to 5 minute stride.
Or maybe not turnstiles for consistency, because the one ticket or one pass works on the metro, the tram, the trolley and the bus. You really can't put turnstiles at ever tram, trolley and bus stop and that’s the overwhelming majority of the use of the tickets.
Then what would we do with the old folks (no ticket required for 65+) and the Ukrainians (no ticket required)? Now we would have to print free tickets to use at a turnstile? Naaaaaa.
How do the German speaking countries handle the free passes for the old folks and refugees?
For the metro I would guess (just a guess) that the labor involved in spot checks is cheaper than buying and maintaining turnstiles. That and the fines the tourists pay for not buying tickets is a great subsidy to the system.
We have an app too, BudapestGo, if you buy a single ticket you must electronically validate it when you get on (there is a QR code at the door); again, not used by many locals. With the multiple day passes, you just show your phone when asked (rarely).
By the way, th co-inventor of the turnstile had a German surname: Trubenbach So that explains a lot.