To me, rougelite games are defined by the following characteristics:

• Random world.
• Random challenges.
• Random loot/progression within a run.
• Permadeath within a run.
• Relatively short runs.
• You lose more runs than you win.
• Slow progression across runs, requiring you to play the game over and over again to unlock new content.

None of that is appealing to me. I like worlds that are worth committing to long term memory. I like challenges and loot to be balanced, and I like being able to plan ahead using the foreknowledge of what the challenges and loot are going to be. I like being able to see the good end of the game on my first run. I like long, epic games. I like putting the game away after my first run and starting a new game.

I never played Binding of Isaac, and based on its gross-out themes, I probably never will. I’m barely aware of Slay the Spire, although I’ve heard the name before. But I think I’ve played enough roguelikes and roguelites (ADOM, FTL, Dicey Dungeons, Darkest Dungeon) to know that they’re not my cup of tea.

The one roguelite I played recently that I actually enjoyed and played all the way to the end (of the plot, without all unlocks) was Desktop Dungeons. I liked how it balanced its random map generation with the most non-random combat system ever: I could calculate exactly how each battle would end, down to the last hit point, before even starting the fight.

Paradox is a publisher, not a developer. You would need to be more specific.

You can't chose your Steam tags, like you can on Itch. The players will decide. And the most popular ones will appear on top. One can even see how some tags change over the years in relevancy to a topic. And one can see that certain tags are often applied wrong. If enough people do it, the meaning even changes.

Your described game loop sounds like Darkest Dungeon - without the eldritch horror bits. Darkest Dungeon is not a rogue-like, nor is it a strategy game. It's not a rogue-like, because the central game loop is not the "run", but the campaign around the runs. You can still say, it has rogue-like elements. But it's not at core a rogue-like. They made Darkest Dungeon 2 a rogue-lite. There is no campaign, only runs. And you need to unlock everything, by failing. And yeah, it is as bad as it sounds. DD1 is a masterpiece. DD2 is ... not. And it's an insult to all fans of DD1. But at core that game is a rogue-lite. And it's lite and not like, because failing and unlocking things to be better next time is by design. You can play DD1 without ever restarting. Infact, restarting does not give you anything iirc. People do not use those words strictly, but by definition a rogue-like is not a rogue-lite and vice versa, yet many games have both those tags.

The loop is basically a party rpg. Hire people, improve, go on missions, rinse repeat. Technically, it is two loops. An inner and an outer loop. One can think about the inner loop as rogue-like. And the outer loop as a rpg with a bit of base building. And if you are unsure, if I was talking about Darkest Dungeon or your game, well, ... ;-)

A bit closer to your King type mechanic might be  Vambrace: Cold Soul. An underrated game. The inner loop is the rogue-like elements of Darkest Dungeon. But you do have a hero character. And the outer loop is rpg adventure style. Both games have more going on than their mechanics. For Darkest Dungeon it's the atmosphere and eldrith horror theme. For Vambrace it's the story.

So what your game's tags will be, will also largely depend on the surroundings. People tag more than the game mechanic.