Oh, that would be an interesting idea. I've imagined a competitive horde mode myself, spitballing reasons why Splatoon 1 had improved Octoling models since I never believed they'd become playable, but a Warriors mode like the Tecmo Koei games would be fascinating. I don't think it would happen, though, since it wouldn't work well on the existing maps.
And just one AI "queen" to protect? No thank you. I hate escort missions. Rainmaker is just right, since one person isn't the bearer the whole time, but if it were a permanent assignment, especially if it were an AI, that would be frustrating as heck. Besides, it would be a severe programming challenge. The octolings in single-player aren't very bright; a robust AI would take a lot more doing.
Now, if you wanted the idols to show up in battle, how about this: If your team is the #1 team in League, you get to use a custom avatar for 24 hours in Ranked and Regular. You have a few presets to choose from, but if you own a matching amiibo (Callie, Marie, and whatever others they print), you can use that.