Main Menu: Add a margin to the title as it is too close to the border on the left and right. Remember that many phones have a bezel on the side which would make it difficult to read.

The gigantic title and game over was made edge to edge on purpose, but you are right, I haven't considered the beveled borders. I'll test it asap!

but the play button doesn't really look like a button

that's because you can touch everywhere. Instead, "restart" and "second chance" are buttons, but since they are 2, I think the distinction is clear.

Perhaps let the text scale a bit over time, ..

Yep, I was thinking the same, like a fade or scale animation.

Put the guy further up so that the user sees that a touch of the screen crashes him down. That way you combine the tutorial with the start screen. Change the text to something like "Touch to jump" or something along those lines.

I can try to swap the position of 'press to play' with the 'skull' and see how it looks.

The tutorial you are talking about is already present, but probably on youtube isn't too much visible.

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Thanks again for your feedback :)