Bingo Connect6

Could p1 advantage be mitigated if the center point was a neutral color that counted for both players?

Alternately, p1 plays two stones on their first turn (black and neutral) and p2 chooses color.

I had a shower thought today to take the free space from bingo. It doesn't work for every game, but might help for some games.

How big is the advantage for p1? It seems like strong players win regardless of color?
My impression was that at worst p2 can play a defensive opening quite successfully, even at 2100-2300 ELO. At best, the game seems relatively balanced!

The suspicion is that p1 wins with perfect play, so it's something that I've been thinking about trying to solve for competitive balance.

I tested out the wild-stone idea a little more after making this post and did not enjoy the gameplay - far too offensive from the start. The game still most likely is better served with a Swap2-style pie rule at high levels.

I had designed variants of Gomoku long ago, including forbidden points, shared points, and points that count as half pieces, which significantly increase the fun and balance but almost always transform it into a significantly different game strategically.

For example this one, Field Gomoku:
0. Basic rules: follow Freestyle Gomoku or Renju.
1. Opening phase: The first stone is in the center of the board. The first player designates 3 six-point spots on the board. The second player may swap.
2. Six-point spots: Stones placed on these positions require six-in-a-row to win.
3. Endgame: If no victory is achieved, the player who occupies more six-point spots wins.
(This is a relatively balanced variant designed to prevent current AI cheating while staying closest to the original gameplay)

"p1 plays two stones on their first turn" would make the opening too close to the edge, and it would become more difficult to find a balanced position. Most importantly, this still requires a swap, losing some of the unique beauty inherent to C6.