Making a new 3v3 map wouldn't solve anything. It might draw some attention temporarily (as you said, because it's new), but that's just a stopgap. Making 3v3 part of the esports series, having events and/or Missions, or at least regular updates would be much more impactful.
At present, the mode itself is decent. Sure, there's various issues (e.g. Support meta), but that's true for every map. The main problems are just that there's no change or variation and little incentive to play TT. SR players complain when certain power picks remain strong for a month or two. Meanwhile, TT's meta hasn't really changed in a year or two. Furthermore, there's not much reason to even play TT. Aside from Challenger rewards or the Gold chroma, Treeline doesn't offer anything. It's explicitly excluded from Mission progress (5v5 PvP games only), and Champion Mastery tokens still aren't available there.
I can't tell if Riot did or didn't learn from Dominion because this is frustratingly similar. "Give people no reason to play a game mode, and then act disappointed that no one plays that mode." It's even more frustrating that the gameplay plans video mentions refining new game modes, yet has no mention of actually supporting League's existing game modes.