You have all that with Gacha pulls and battle system.
In fact, I would describe this as an idle game, not a gacha game. In other words, a gacha-themed Cookie Clicker reskin. The emphasis is on the gacha, but in a real gacha game, rolling gacha occupies <1% of the playtime, and this doesn't adequately capture how the gacha model intersects with real gameplay mechanics.
Gacha is not related to battle systems at all. wordpad was referencing the random awards from the battle system as another funnel.
Constantly trying to no-true-scotman what Gacha means, is not definitive or compelling.
The specific type of battle system is not inherent to a gacha, but having a gameplay system of at least moderate complexity is. The one thing that virtually all gacha games share is that they have a large number of distinct game pieces of varying rarity. This lends itself specifically to deckbuilding/teambuilding. I've played dozens of gacha games and they all have teambuilding elements. I'm sure you can find some exception that proves the rule, but it is the basis for the genre, insofar as it is one. If you wanted to actually boil a gacha game down to its abstract essence to showcase what playing a gacha game is like to someone who had never played a gacha game before, you could not possibly give them an accurate experience without it.