Hola! I played for about 10 games and I think you’ve got the “just one more run” feeling just right, congrats on that!
However, for this kind of games it’s very important to be able to get into the next run as fast and easy as possible and the current flow is frustrating. Take a look at the incredibly successful Super Hexagon, even in its arcade mode, for a really good example of how to do this effectively (it takes literally milliseconds to jump from one run to the next).
First, I would suggest that the end game screen has instant restart actions in the middle and keeps the “back to main menu” one in a little button in a corner.
Second, it really needs to remember initials, you can’t be asking for that after every single run! It doesn’t need to keep them between sessions, but when I last entered them 15 seconds ago, they have to persist. (I mean you can still bring the screen up to enter them, but pre-select the letters from the previous run).
And third, the loading animation is unnecessary and slows down the time to get back which makes me want to quit, a little more after each game. I suppose it’s not actually loading anything anyway. The truth is that it feels and looks nice and matches the aesthetic of the game, definitely keep it for when loading into the game from the menu at the start of a play session, but don’t include it in the direct restart action I mentioned in the first point—or find a way to make it last less than 500ms at most if it can look good that fast.
These might sound like small details, but they can make the difference between somebody playing for an hour straight and leaving satisfied or getting frustrated and closing the game after only a few tries.
Wish you luck!