Presentation and help with sprite on/off

It is long time since I've coded in Basic so mine is just a guess
Try adding a delay between line 570 and 580
Yes, add
575 for i=0 to 10:next
to make it work

I almost never code in Basic, but according to MRC documentation, SPRITE ON|OFF instruction seem to be asynchronous. If it's the case (detection not occur during your loop but during the interruption handler I guess), there is great chance that SPRITE is ON at the moment of check.
What you are trying to do looks like the OR-ed sprite management for the MSX2 screen modes.
That said, it's possible that Basic does magic that I don't understand ^^

Thanks a lot for the explanation. So waiting for a frame (a change to TIME) seems to be the correct choice, if I understand correctly.

Yes. It is. In addition, if you put the put sprite commands just after a VDP frame has ended (and if your code is fast enough, i.e. all the parameters are already computed and stored in variables), the sprites will be set in VRAM before the raster has reached them. This implies that collisions will be computed on the coordinates of the current iteration of your game loop and you can avoid the SPRITE ON/SPRITE OFF trick.

https://msxpen.com/codes/-MuGFFMX5_X1KuZc4tXd

Sadly the interpreter is slow and can introduce delays so you cannot be always sure of its timings

The difference has to be in the the timing and in the duration of the code that sets the sprites
Anyway, note that this line
780 IF TIME<2 GOTO 780 ELSE TIME=0
waits for frame ending