I'm removing from inventory FE: special edition now, which says it includes "Conquest". Your description sounds great, as less story/cutscenes and more "classic strats" is exactly what I want, with greater ability to customize units/squads than what Awakening is providing me.
Actually customization is
worse in Conquest. The latter is superior in the tactical aspect. See, old Fire Emblems didn't focus that much in the RPG aspects, they were more like tactics games, there wasn't as much customization and there were no procedural battles (except in Sacred Stones) to develop your team and focus on that. The games were very tightly balanced to offer challenging tactical combat, and to give the player too many options plus grinding would have made the games harder to balance.
A lot of FE fans complained about Awakening precisely because it broke with that tradition and went for a more RPG route with more emphasis on squad customization. I personally like both the old games and Awakening, and if you play Awakening in Lunatic and + you still get a classic hardcore FE tactical experience with no grinding (IMO actually better than most except for New Mystery of the Emblem and Conquest at their highest difficulty).
If you want more like Awakening, play Birthright, Revelations, or Echoes. Revelations is the best of these, so start there. Conquest is more like old-school FE. Don't get me wrong, Conquest has a good character system based on the newer games, but due to the lack of procedural battles, building up your team is trickier and requires you to do battles in a certain specific way. It is a more challenging but less freeform game overall. Both types of FE are good fun at the end of the day, so try them yourself and see what you make of it.
As for the story, in all of these games you can just skip the cutscenes easily, so it doesn't get intrusive if you don't want it to. The only case in which I recommend not skipping it is Echoes, which has an okay story that adds to the game.
It is a different case from Shadowrun since SR games are mostly of the same type, while Fire Emblem games have two "sub-genres", for the lack of a better term. There is the powergamer FE and the tactician FE. The best of the former are Awakening and Revelations, and the best of the latter are Conquest, New Mystery of the Emblem, Thracia 776, and Radiant Dawn.