I have never heard the term "rote" before.
If rote is a less complex way to pick up music I'm all for it. If rote means one learns from broad concepts and is allowed some personal flexibility. I love the concept. Chord charts, number systems etc. I guess these would all fall into that category.
If this is true then, critical must be mostly the opposite. Diatonics, chord inversions, what things mesh well with other things musically.
If you're asking for examples on critical, probably almost anything classical, and this kind of thing lends itself well to midi. Maybe the composer feels there is an emotional element in their music but it is all distilled down into notes on a page and parts in exact keys and placements. Learning from this persuasion is all about learning your scales, chords, keys and musical relationships on paper.Lots of emphasis on technique. How things are supposed to sound by the rules.
OTOH an example of rote might be a few pickers sitting on a porch in West Virginia somewhere. Neither of them had any "trainin". Other than maybe someone in the family said that this is how you do this and that and they picked it up. No one would say these guys can't play, but if a classical piano player on vacation came along and wanted to join in he had better be a pretty good ear player too.
Years ago when I took trumpet I was told how I was supposed to hold the instrument. Then I saw Miles Davis breaking every rule in the book but he was famous and holding the trumpet the wrong way.He could hit notes I never dreamed of.
I certainly fall into the rote direction. I don't like all those rules. I have had some formal training but I seldom use it. I played a special number on piano this past week that I didn't really know what I was going to do until I did it. I played with something for about a half hour before I went on. I took something structured and added some stuff to it here and there. It went off without a hitch. I took a song written in a major key and I made an intro in the same format in a minor key, then I merged it into the major key and added some other stuff to that. It all seemed to fit together. The week before that I played a song using a looper and an acoustic guitar. I added a rhythm guitar part, a bass part and some vocal backups on the looper. I switched certain parts on and off and added a harmonica part. I learned guitar by "rote". Still have a lot to learn but it's fun. We had a pretty full house that day and I got a standing O. One of the few I have ever gotten. I really wish I could concentrate long enough to let someone teach me, but I usually end up learning what I can and learning the rest by trial and error.
If I'm not playing right that's ok. I'm having a lot of fun doing it the wrong way.