Of course you can make everything yourself if you want to. Game engines aren't magic. There is nothing they can do which you couldn't program on your own if you were willing to invest a lot of time and effort. But the question is is it worth it?
Using a premade engine means you have already solved a lot of the common problems every game needs to solve, like asset loading, rendering, collision detection, UI framework and many more. Doing all that stuff on your own requires a lot of studying algorithms and APIs, a lot of programming and testing, and in the end you will just have ended up reinventing the wheel.
But if you use a game engine, all of that drudge work is already done and you can put all your time into what actually requires creativity: gameplay and content.
Sometimes you have game concepts which are so unique that they require special technical implementations no common engine can handle. For example, Minecraft with its up to 18 million of destructible blocks in the scene at once would never work in an engine not optimized for that. In such a case you need to invest the time to develop your own engine. But that's rather the exception than the norm.