Exactly, that's the point I was getting at: so when you see a pattern, how do you approach it? Here's an example: I see multiple patterns but how did you go about solving this mate in 2? (Because only one pattern eventually is relevant.) Personally I couldn't figure it out pattern-wise so I used raw calculating power again.

Well I've played chess for a long time and I'm rated above 2000 (OTB) and I've never heard of dovetails and swallowtaills so make what you will of that...
Seriously speaking though, we practice tactics in order to easily recognize common patterns during our calculations in an actual game. Pattern recognition is mostly an intuitive process instead of something you have to think consciously. It just takes a lot playing and lot of tactis exercises to get enough repetitions. I'm afraid there are no shortcuts.