I think it following the grid would be better yea, maybe you could just remove the possibility of falling into the pits. (You already have the R restart if we get stuck, also I think you could make the R button works twice as fast lol).
Also I feel like the pushes are sooo slow. I think it would be nicer if you could push faster? (Though in the later levels I understood why the character doesnt just move while it pushes).
Could display the numbers of level left too.
Is this power creep...?
I'm not a fan of those levels with lot of superfluous stuff. Like this one too with the extra boxes and extra destroyable tiles 
Nice little game otherwise!