Speaking ASA
The classification process wasn't based on skill rankings, but purpose of the team. An A team was that of a travel team that could fill their roster with no geographical restrictions and a B team was a local rec team that to play in the tournament, all roster players were required to play in the same league/organization. An A team could not play down, but a B team could play up.
And that makes sense. That's like high school sports, where teams are classified by their characteristics (primarily enrollment, but sometimes public/private). There is legitimacy in the idea that 'this is a contest between teams playing within these rules.' But, as someone noted above, B teams might well be far better than some A teams, just as some Class AA high school teams might pound some AAAA teams. So even within those A/B divisions of that ASA has set up, you're going to have mismatches that might be managed well with a ranking system.
And speaking of ASA, I sure wish more ASA tournament directors would use the software or programs available for getting up tournament results on ASA websites. I hear there is some kind of fee, which might discourage some from using it. I don't know. But I sure would like to be able to find results/scores of ASA tournaments, especially qualifiers and nationals, a lot more often and faster than I currently can.