Interesting idea.
I think that this game has a good idea, but it somewhat falls apart within its execution. Namely I think the spikes are just too difficult to time yourself with, given their count only gives you one correct permutation of timings that will work, so you are then left to just sit around and wait till it happens. Giving them a longer sequence would provide the same puzzle idea of 'Pass the spikes on the beat when they are down', but prevent having to do it perfectly.
The levels where you are also just pushing boxes, while more fun due to the lack of spikes, don't really use the mechanic in any interesting ways, since you are just simply waiting to move.
I think this game would maybe benefit from being a top down style, and giving a 4 direction movement. And making most of the enemies / obstacles move in a predetermined pattern. Since then it will be easier for players to predict where enemies will be and track their movements and timings.
There are also some other QOL things that could be added, that would improve the feel of the game for players. The first one that comes to mind is letting players read the opening text before you start any countdowns, otherwise they will likely die trying to read the text and understand how the game works. This method should also be applied to all levels, in that the timers don't start until you do a initial input, so players can scope out an idea of how they will traverse.
I like the idea of timing the game with the music, and it brings some of the 'feel' of the game. But I think you could've taken a different approach and map out the bpm of the song on a timer of some sort within the game, and then you wouldn't need to keep restarting the song to keep it in sync. Since being able to listen further into the song would make it feel less repetitive.
You could definitely also get fancy with the music, and have it have moments of multiple beats in quick succession. So the player would have to quickly plan out a route, especially if that section of the song has a memorable part right before it.
The art is very charming. And having a simple intro cutscene adds a lot to the character of the game.
There is good potential here, and I think it would just require some reworking on the mechanics :)