That sounds easy but it reality would be way more difficult to put in place. The reason it works in soccer is that there is an overall governing body, then a regional and local governing body, all chartered to do the same thing. That is a significant amount of regulation. For it to work you can't do it halfway. There would be a lot of leagues that frankly would not see any benefit and therefore next to impossible to While it would be nice to have players registered, the overall concerted effort to do that would be huge.
The only possible exception to this would be US lacrosse. Since most of the leagues and almost all of the tournaments have moved to use US lacrosse's insurance benefit, there is a clear connection with US lacrosse number in which as part of the sign up each player could receive a player pass. Not perfect and bad intentioned people could work around but for the most part that could work.