I played From the Core, good stuff.
It was enjoyable to figure out how to use the apparently simple mechanics we start with at the beginning to learn how to do more and more with it. I actually wish there wasn't any hint, because it felt much better when I figured out a mechanic by myself rather than when it was explained to me.
My main criticism is toward the controls. I suppose it was made to feel like the old style metroid games (and maybe also to make sure the player can easily aim straight down when jumping?), but being able to only shoot in 8 directions was a bit annoying. It makes aiming extra hard in some situations, in the wrong way imo. The character movements felt heavy and slow, and the charge shot also felt pretty long.
I couldn't play with the controller because my mouse was conflicting with it, and it was hard to place the mouse relative to the character to aim correctly. Maybe use the mouse position relative to the center of the screen, that way when doing a long jump for example, we don't have to move the mouse along with the character.
I'm not a huge fan of the aesthetic, but I guess it's an homage and that will really be a personal preference :)
I definitely felt the sense of progression without power ups that you are talking about, so great job on that!