There are a lot of different ways to start this game, and all of them are valid, including the official original rules.
I have a feeling the answer is that there is too strong of a first player advantage in the standard rules to let it count for Elo, BUT
There are lots of games with strong first player advantage, including Chess, which still use Elo just fine because over many many games it balances out. You get to go first about half the time, and you win some and lose some, but overall, if you are better, you have a higher Elo, even if sometimes you lose to a first player advantage.
I have a feeling the answer is that there is too strong of a first player advantage in the standard rules to let it count for Elo, BUT
There are lots of games with strong first player advantage, including Chess, which still use Elo just fine because over many many games it balances out. You get to go first about half the time, and you win some and lose some, but overall, if you are better, you have a higher Elo, even if sometimes you lose to a first player advantage.