Using the ABBA system as well as the toss of a coin is fairer.
What is it you don't like about ABBA?
Apart from their music, obviously
It's confusing for supporters and above all it is completely unnecessary. Statistics may well show that the team taking first is more (or less) successful, but as you can't recreate the exact circumstances of a shootout a second time with only the order reversed, you can't make the case that any "disadvantage" exists anywhere other than in the players' minds.
A coin-toss has a 50/50 chance of any one of two possible outcomes. Therefore it is the fairest way to decide things like who kicks off, or who takes the first penalty.
A penalty kick is striking the dead ball towards goal from 12 yards against a goalkeeper who (should be) on the goal line. Everyone here understands that, it is a simple concept.
If there are psychological factors which affect a kicker's ability to successfully take a penalty, it is up to that player (or their coach/sports psychologist) to deal with that. I repeat, any psychological factor on the part of the players is not the responsibility of the laws of the game.
That pressure is part of the game, and the team which is able to cope with that pressure better will be more successful regardless of the order the teams take their penalties in.