As an Alabama fan and former player, I hope to God that Saban and Byrne say "NO." It took years for Alabama to put player's names on the jerseys. We don't identify individual accomplishments (helmet stickers). We don't identify captains with the "C" on their jerseys (permanent team captains aren't picked until the end of the year). We don't retire jersey numbers. We don't have any type of ring of honor. Everything Alabama does is about the team. The only thing Alabama does is recognize players who have graduated (which is an SEC initiative, not an Alabama initiative).
Saying no doesn't mean that social causes aren't addressed. I don't think you would have 100% agreement by players on which social cause to support. The NBA is allowing each individual to pick their cause (here's a list of which NBA player will display which social justice message - Season restart: Team-by-team social justice messages on jerseys | NBA.com). Individual messages do not unify a team.
Just because it's allowed doesn't mean it should be done.