The times to get the stars are only present at the end of each level (in the tab that is opened when collecting the microchip) but it would be more correct to include them in the selection menu as well.
I agree that the purpose of the stars is indeed ephemeral, a player can decide to strive to get them all or ignore them not interested in the challenge. Giving a reward for achieving a number of stars (such as unlocking levels, dialogue, endings, cosmetics, or in-game currency) is certainly a clever way to entertain players who are not solely interested in the challenge.
The puzzle approach I usually rule out since these gamejams are always saturated with such games, but the idea of providing a speed boost based on the scaled-down axis is something that would definitely have interesting applications.
The idea I started from here was actually was to create a very basic version of Neon White, then make the speed of level completion a priority challenge.
One design (or rather geometric and engine) problem I had was that resizing on the z-axis could virtually negate the need to have to choose other types of resizing. I solved it by reducing the speed when the hole was mismatched and limiting the ability to resize on that axis by level. Initially, though, I had solved it by making that transformation a little bit thicker in the x and y axes so that it still didn't go through the other holes, but capturing the velocity caused Unity to make the colliders interpenetrate anyway. If this solution had worked, the levels would definitely have been geared toward a more puzzle approach.