I think it should permanently be Europe v North America. That would have worked in most decades:
80's (not all playing at once in some cases but would be the mainstays of the team):
Europe: Willander, Lendl, Becker, Borg, Edberg, Kriek
NA: McEnroe, Connors, Chang, Courier
90's
Europe: Edberg, Becker, Kafelnikov, Bruguera, Moya, Muster
NA: Sampras, Agassi, Chang, Courier
2000's
Europe: Federer, Nadal, Djokovic, Murray, Safin,
NA: Sampras, Agassi, Roddick, Isner
2010's
Europe: Federer, Nadal, Djokovic, Murray, Wawrinka, Cilic
NA: Raonic, Isner, Querry, Sock, El Shapo
The North American team would have been weak through the 2000's and early to mid 2010's but are probably well placed with Canada possibly leadning them back once the big 4 move on.
The only players of some note that would have missed out on playing Laver Cup had the North American team was a World team during the above time period that I can think of are:
Cash, Gomez, Rafter, Hewitt, Kuertin, Gaudio, Nalbandian & Del Potro that is only 8 players since the 80's - even allowing for a couple that I may have forgotten, let's say 10 or 11 players missed out means that the very best of the tennis world is Europe v North America so the Laver Cup may as well just play those two and not allow the 'World Team' to become invincible when the balance of power swings back to North America and then have their team topped up with the world's best outside of Europe also.