Carrots often have dirt caked on the outside that's hard to get off with just water, so peeling is a good way to help with that.
The peel has the healthy bits
Sort of, but not really. The nutrients of a carrot may be slightly more concentrated in the skin, but all layers of a carrot contain those nutrients. You're not depriving yourself of an appreciable amount of nutrients by peeling a carrot.
I ended up at the practice after I first started cooking for myself and didn't think to do this and wondered why the carrots were so unpleasant. The peel is just too... carrotty. It's just super intense carrot taste to the point of unpleasantness, also even with a good wash it kind of tastes like dirt. I only really like it when it's those little carrots sometimes referred to as 'dutch carrots' and they're roasted so you get some blackened char on that skin.