DA is mostly vanilla-balanced under AI control, but in player hands can be extremely powerful as mentioned above.
Different factions have different strengths, where DA has very strong and a good variety of Wanzers/fighters, good missiles, and can have somewhat beefy capital ships. Whereas BRDY focuses on fragile and fairly specialized ships that have high maneuverability, decent flux management, and powerful ship special abilities. Very different learning curves between factions.
I think part of the reason you're having issues now is that your playstyle seems very tailored to relying on "wanzer swarm of death (wanzers are slow and VERY tough fighters) + missiles everywhere", so you were playing to the strengths of DA. Also fighters in general are quite powerful: they're essentially flying light weapon, PD, and missile platforms that can regenerate.
Since you weren't actually piloting any ships before (just giving them orders via the tactical map), you're probably not used to piloting so you need more practice there. You may also need to get more practice outfitting ships with a balance of firepower and defense and figuring out what loadouts and weapon mixes (and what kind of PD / how much PD) work well, rather than being able to throw a mix of missiles and fighters on ships so that the enemy is usually too busy fending them off to actually fight you.
Not every playstyle is for everyone. As you've seen on the forums, both flagship solo-ing with a minimal support fleet and running a larger fleet while piloting your flagship are viable tactics. E.g. just because I can't personally flagship solo enemy fleets doesn't mean it's not viable at all
There's nothing wrong with playing only carrier-focused if you don't enjoy anything else, although that does mean you will be limited with what factions you can play as (some don't have as powerful fighters, or have very few fighters which mostly just fill support roles). As mentioned, DM is another very powerful fighter-based faction you may enjoy (however that one is generally agreed to be overpowered when used by the player, and can be a harder AI faction to fight against).