I think that unrated puzzles solved by a user should really count and be displayed on the profile page. I often solve unrated and I would love to see the real amount of effort I put into solving puzzles in general rather than rated puzzles only. For example, according to the stats on my page I solved 18 or so puzzles, while in fact in might be around 70. It might seem insignificant, but I think a lot of people would take pride in seeing that they solved a lot of puzzles, while not counting unrated solved puzzles is kind of discouraging.
There is a point to be made that the number of puzzles displayed might be misleading, because one might climb to lets say 2000 rating in puzzles and then farm unrated easiest puzzles to be perceived as a solid player, but I doubt that this is a concern for the community and I just dont see how it would be harmful in the grand scheme of things. Puzzles are not a competiton, right? Well some players might perceive it as such but come on; Even if people are gonna start comparing number of solved puzzles it still benefits the community
Why not solve rated puzzles if you care about the count? There's already a very simple distinction between caring and not caring about stats in both puzzles and games, and that is rated vs casual, but we keep seeing suggestions to make casual puzzles/games mean more for some reason.
I dont like solving rated puzzles because you might pour in time and effort and still end up losing rating and it doesnt make sense to me in termes of motivation. I dont think its a good thing to fall in the trap of chasing a rating while chasing a number of solved puzzles is totally healthy because it adequately describes your aspirations in terms of getting better.