If you transferred everything it should have kept track of the total playtime for each game. E.g. if I put Breath of the Wild in my OLED and use it, it'll show the correct playtime in my recently played list.
The problem is, of course, there's no easy way to see that other than loading every game on your new Switch and then checking the recently played list. For whatever reason, it doesn't count that playtime when ordering your games by hours played in the software menu (I think because that is total hours play per game on the system regardless of user, and the figure in recently played is however many hours an individual user has played a game).
But basically: Nintendo should have an Activity Log on Switch. The system seems to keep tabs on the data, there's just no easy of way of accessing it.