Dominions shouldn't be lumped with "wargames" just because of how it looks or seems to work. It's far more complex than you seem to give it credit for, and most of that complexity isn't on the battlefield. It has far more in common with the likes of Master of Magic - a game officially defined as a "fantasy turn-based strategy 4X game" - than any of the examples you've provided. Just because all combat is automatic, does not make it less of a 4X strategy, just as Master of Magic or any other 4X wouldn't be any less a strategy game if you used "autoresolve" for every combat - indeed Dominions is even more of a strategy game than that, as it gives you a degree of control over how a battle "autoresolves" that no other 4X strategy game does.
I don't know. Not that you can't play MoM on a strategic level, auto-resolving everything, but so much of the power and beauty of the game was from tactical combat-level shennanigans. It's not the only way to play, but it was the way it was meant to be played for it to be a fulfilling experience.
I see Dominions as more like Stars! (an old space "4x"). In that, the stars you explore, colonize, defend, capture, etc, really were just a part of the machine of your war economy. Resources were king (with current available minerals determining easily available makeup of fleets), for everything, including research, exploration, defense, and aggressive moves. Combat was similarly automatic, but composition and particular ship components had huge impacts on how battles played out.
But the strategic layer wasn't *just* about throwing more resources and minerals at the problem. Especially in MP. Race type, race design, galaxy parameters, overall strategy, colonization spread/focus, fleet type/specialization/add-ons, extra defensive/attack layers (minefields/mass packet launchers/stargates/scanning ships/cloaking) all had a massive part to play in the outcome of any match. Not to mention the diplomatic shenanigans going on behind the scenes in anything greater than a 1v1, because of the different needs and acceptable environments of different racial builds.
And likewise to Dominion games, it became immensely enjoyable, immersive, strategic and tactical in multiplayer games, to a level that you'd never really ever realize if you were just playing SP against the AI, as it was essentially at war with you by default.
Was Stars! a 4X? Yes. So is Dominions 5. But they're ones that don't lose out from not having a specific "player controlled tactical battle" section. I feel MoM does lose a lot if you don't utilize that part of the game. Where as those two are "tactical", but in a different way.
Playing cat and mouse with minelayers to sweepers with counter-incursion patrols in place, as you try and thread your cloaked incursion fleet through their scanner zones is certainly "tactical" in Stars! But you got there by overarching "strategy". And you'll never give your fleets an order when combat does begin other than what they were already set to when you hit "end turn". But you'll definitely have "tactically" moved them, with the "strategic" layer of the game being of a far greater scope.
Dominions is kind of like this too. But not like MoM, despite the genre similarities. In many respects it actually loses some of both these tactical layers for a more "meta/matchup-path" style gameplay, but is still "tactical" in just how you move and match-up those forces against your foes. Stars! was far simpler than Dom5, but had a far more fluid tactical layer in movement, even though they really are just about "lots of army thrown against the enemy due to my BS economy" on the strategic layer. But how do you get there? And how will it be used? You'll be using your strategy to inform your tactics, and vica-versa. If you're any good and know what you're doing, that is