It would be interesting to see if there is anything which would let black start on equal terms.
This would theoretically allow black to start on equal terms.
One you play as black and the other white.
"Basque chess - A chess competition in which the players simultaneously play each other two games on two boards, each playing White on one and Black on the other. There is a clock at both boards. It removes the bonus in mini-matches of playing White first. Basque chess was first played in the 2012 Donostia Chess Festival in the Basque Country, Spain. Also called Basque System." https://en.wikipedia.org/wiki/Glossary_of_chess
This story on the event says it was originally Bronstein's idea. https://www.bostonglobe.com/arts/2012/01/23/chess-notes/0GSY...
Was the game fair for both players?
Or more, based on https://lichess.org/analysis you'd have to ban a couple more moves like a4/h4, but you could opt to restrict picking first moves within Stockfish range at some depth of -0.1 to 0.1 (looks like c4 is enough to be 0.1 at depth 47)
This may increase the draw rate
If you want to keep engine analysis out of the rules, you can go the "evenly split candy between siblings" route: after White's first move & before Black's response, Black may elect to trade positions
It also means White still gets the edge of being able to prepare for the first move vs their opponent, ie they may've studied 1.Nc3 lines much deeper in preparation, whereas currently White gets the edge of choosing the first move & it's advantage
ie White gets to decide whether they prefer their e4 prep or d4 prep against their opponent's prep as Black, so if White doesn't want to play against their opponent's Sicilian they play d4. This is a theoretically moot but real advantage. See Fischer's performance with c4: https://www.chessgames.com/perl/explorer?pid=19233&side=whit...
See also how well 1.g3 or 1.b3 have performed while Stockfish gives both a solid 0 eval
In short, while I started with Yes, I mean No. Because b3/g3 are equal enough that White gets an edge by deciding whether the game will be a g3 or a b3 game
Sure. Why would that be the goal though?
Many of them do reduce the difference between black and white, by increasing the number of draws, but that's not generally seen as desirable.