I agree that in principle one ought to be able to do without thinking explicitly about logic, algebra, etc... but is that how it really goes in practice?
It's not, and I think that's why some people take a long time "getting" monads. Lots of tutorials online spend a lot of time talking about category theory to introduce monads. You can see the opposite view in things like this: http://dev.stephendiehl.com/hask/#eightfold-path-to-monad-sa....