Well one thing that came to mind while I was thinking about the name of the game and the rules was the mobile game snake.
The snake has a head and a tail, and you kind of have to keep moving in the direction you start with, and when you change it can only be the head that changes the direction.
In that sense if you made it like you have to place a stone adjacent to your last placed stone then there’s some funny odd/even effects that can happen which change which path you might fill in the board, or might not be able to fill at all.
Like if black follows the blue path vertically, they can run out of moves quicker than if they swap to the green path, or later to yellow say.
White also runs into some issues with their edge, probably to do with leaving a gap at D5.
I’m not sure it fixes the game, but it introduces slight complications.
I’ve seen some games try to ban an obviously good/unique first move, so one could try ban tengen as a first move if that was winning. Or similarly other games introduce a pie rule placing one stone of each color, or two stones etc, and then letting the other player pick which side they want to play.
Alternatively I’ve also seen some games like Abalone change the setup of the game from the original to a tournament setup to make the game less drawish.
So there’s different avenues to explore to try and balance a game that isn’t immediately solvable.
If you ban tengen as a first move, say black plays triangle here and white plays tengen
Black immediately has a direction choice to make, and because of the “new” snake rule white can choose the opposite direction, and black will be a bit slow to circle back to try and block.
As in like this
and this
You’d have to count the spaces and also find the longest paths you can make along edges and things.
It makes it at least a bit more difficult to understand what might happen, especially if there’s now some complications with the pathing around the edges.
Edit: continued
The obvious thing to check is white blocking black by following and copying but there’s some subtleties
Black 22 seems like a mistake because it creates an issue around the X’s.
While intentionally leaving a dame between black and white seems better for pathing (while before, previous position, it might’ve been a bad thing)
And I, off the top of my head, don’t know what the strategy is to win as white. It might be possible to figure out though.
But even these aren’t necessarily forced lines, they just kind of make sense from a maximising space sort of idea → don’t let yourself get boxed in and run into issues having no moves left.