The very first Half-Life was already using one of the Quake engines. As was Hexen.
The first Deus Ex was using the Unreal engine.
That was all probably ten years before Unity.
These engines being tailored-made for a specific type of game, people wanting to do something else had to create their own engine, often in C++, sometimes in Python or Lua, reinventing the wheel most of the time. The barrier for entry being so high, game designers had to team with coders, but their end-goal not being always aligned (creating the game vs learning how game engines are coded) this lead to the demise of many projects.
With Unity, a general purpose game engine became available for a very low price (in particular compared to Flash). Many game designers saw it as a way to work on prototype without needing to recruit a team of coders. And for many coders, it was a way to try their hand at game design, while being able to push the engine further using C#.
And even then, CryEngine, Unreal and Source are all engines centered on a very specific kind of game (fps style). There was no serious general-purpose game engine before that, that devs actually used.