Valve already has a game engine you can use – Source – although outside of their own games, it’s not really popular. Otherwise I think it’s moreso that making a good general gaming engine is hard. Like, really hard. If Valve tried to compete with, say, Unreal or Unity, (especially with their relatively small team) it’d more likely than not have no chance at all. They’d need a LOT more manpower, a massive budget, and to hope that they actually make something quality enough to actually be a viable alternative. Even then, though, it doesn’t have the 2 decades of content and design that Unreal and Unity have, which is pretty important. Although I suppose Source does have a lot of user-generated content.
It’d be a gargantuan investment, a massive risk that has a high likelihood of not turning out well, and even if it were successful it would likely take many years if not over a decade to actually see the benefit of it.
There’s a good reason most games use an extremely small amount of engines, either that or their own in-house engines. It’s a monumental task to make a great, easy-to-use, generic engine like the ones currently on the market.
IMO Valve trying to enter the game engine market would just end up being either Godot but worse, or Bevy but worse. It’d be far better if they just created a team to work on a pre-existing open-source engine, although I guess there’s not any money involved in that unless they for some reason used the engine.