SuperScrounge, I'm glad you brought that up. I had the exact same thought myself. Without the All-American characters, DC would be without Flash, Green Lantern, Wonder Woman, Hawkman, Hawkgirl, Black Canary, The Atom, and more. Assuming that DC progresses through the 50's in the exact same way and decides to reboot superhero comics for The Silver Age with a team of seven, you'd have to replace 3 of the big 7. DC would still have Superman, Batman, Aquaman, and The Martian Manhunter. To fill the last three spots, maybe they'd be forced to bring back Johnny Quick, Dr. Fate, and Liberty Belle to make up for The Flash, Green Lantern, and Wonder Woman respectively. They could've also reached out for Spectre, Hourman, or Sandman reboots.
Or... It might have put Green Arrow in the team's line-up from the beginning. That definitely would've made it seem like a new Seven Soldiers team. That could've lead to Shining Knight, Crimson Avenger, Star-Spangled Kid, or Vigilante reboots to fill out the team? Regardless, they would have to use Batman and Superman to bring readers to the team.
This might have also forced DC to create other characters sooner than they did. For instance, Supergirl might have received an earlier debut or, possibly, Zatanna. Maybe the Silver age Batwoman at Batgirl would've stuck around longer to make up for the lack of female heroes. Hell, maybe you would've seen Lois Lane become Superwoman.
So many possibilities!
The big 7 team that I would pick in this scenario would probably be: Superman, Batman, Supergirl, Johnny Quick, Dr. Fate, Aquaman, and The Martian Manhunter. I'd call them The Victory League... or something like that.