No format will work when the players aren't interested enough. Only individual sport that seems to make a team event without money on offer work is golf. All the players there would die to play in the Ryder Cup, still in 2018. Arguably gonna be the most hyped golf event ever next month.
I think the way it could happen for tennis is a big Davis Cup(probably called something else, take the name to the grave and be happy for what it gave) only every other year. A two week thing for 32 countries or so, maybe spread around a bit the first week and then a finals venue for the second week, top-8 or so. Let the singles ranking be the determining factor for what countries qualify.