How valuable is it to learn math deeply?
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Deduction and analogy seem like largely different reasoning processes. I suspect that what you're describing is that by learning the notation and doing enough deductive arguments, the tasks begin to become intuitive, that is, they begin to become analogical and not deductive.
Deductive thinking is conscious, deliberative, and "slow." Analogical and intuitive thinking is unconscious, nondeliberative, and "fast." So you're probably right that by learning to relegate many mathematical tasks to analogical thinking, one increases their efficiency of learning across domains.
This means that as you jump across mathematical problems you start to see that one telescoping argument looks like another, or one proof by contradiction looks like another, just like your brain has assembled an otherwise arbitrary class "tree" by taking many samples of trees across many domains, building up some kind of conditional inference algorithm for recognizing "trees."