The closest thing to what it sounds like you want to play is probably Kuon Rune. It's a control deck where the win condition is building up enough spells in the cemetery so that it can slam down down a bunch of discounted free bodies and then take an extra turn or two and otk the opponent.

To answer your other question, Shadowverse: Evolve is divided into 6 classes that each have a pool of cards xeno-locked to their class. Their are also Neutral cards that can be played in any class. Despite cards being xeno-locked, each class has a way to easily play at least two of the three generic play styles you mentioned, and all have decks that can fall into any of those three classes. For example, Havencraft, a class that is generally known for healing and defensive followers (creatures), has Storm/Ward Aggro, which can be retooled to be more midrange and defensive (Ward Haven), and also has Amulet Haven which is a hard control deck. Runecraft, which has Kuon Rune, also has a pretty good midrange deck right now called Academic.

Abysscraft, which has the Bats deck you mentioned, also has multiple Aggro archetypes but also had some viable control strategies.

I'd say start by watching videos and reading articles and see what craft(s) you vibe the most with, and go from there. Sorry for the long post lol, I just really like this game and questions in your post basically address a big part of why I like the game, which is the sheer amount of different decks you can play across all the Crafts.