mettelus
that underlying grid needs to be really close to the notes before quantization or they will shift in a "seemingly haphazard" fashion.
The MIDI file that the OP has provided the link for does not need quantising. I don't know why, but the whole song does not fit into a 4/4 grid, even though it's a 4/4 song.
try to follow this if you can:-
I then selected all of the MIDI tracks and selected the following command, with the now time set to the start of the song:-
Process > Length
That brings up a dialogue box.
In that box, make sure that "Start Times" and "Durations" are ticked.
Set "By" to 80 percent.
This will move all the notes back onto the beat.
I'm off to bed now... I'll call in first thing to see how we've progressed