Now, now, there actually are a few Hungarian dishes made without paprika (I’ve seen them for myself!) but, yes, when I cook Hungarian food, I generally start with onions, lard/oil, paprika. That’s the Hungarian trinity.
Now, paprika shows up as a central ingredient in Hungarian cookbooks, so far as I have found, only fairly recently, like mid-to-late 1800s or so. I found a couple of old Hungarian cookbooks from c. 1820 and c.1850 on Google Books, and very little paprika use could be found. No goulash, no paprikash, nothing like that. There is a dish called “gulyás hús,” which is a precursor, I assume, to goulash, but not quite the same thing. That one did use paprika. Paprika was certainly in Hungary at the time – I believe it was introduced around the 1600s via the Turks, but didn’t gain great popularity until later. Part of it was, I believe, both breeding for less spicy varieties and automation in the seeding and removing of the inner membrane to allow for finer control of heat levels. Paprika is a chile, after all, and its original form was spicy. Now you have many varieties.
ETA: I just noticed I had typed that while eating a nice paprika-laden bowl of goulash(soup).