It seems to be a slight misunderstanding here what a engine actually is. The foundation of a game is always called a engine. Can you build a car without a engine? No. But you can build your own, you don't have to depend on any premade engine. Your engine can be a tiny 100 lines of code or it can be a big beast of hundred thousands lines of code. Either way its a engine. Building big reusable engines is something rather new, ten/fifteen years ago every engine was a very specialized one, specifically build for the one game it was used in.
Building your game from scratch can teach you a lot about the implications of design decisions and I think every game developer should have gone through that process at least once for a small game.
However using a already existing engine will make a game require a lot less work, and why do the work again if it was done once already?
I tend to interpret the slogan "Make games, not engines" as "Make your engine as small as possible" because many programmers tend to work years on the engine for their game, implementing many things they don't really need, rather than the game itself. That way they never finish it.