Let us look at an endgame where White has 8 pawns, and Black has the e/d-pawn missing. How can White win this endgame if both sides play perfectly? I am having a little trouble.
Just move your pawns and your kings up solidly, avoid giving counterplay. At some point you will get the opportunity to get a passed pawn, that you can then usually trade to get several pawns on a wing.
Two short game examples played against stockfish with just king and pawns in initial position while removing black d pawn. First one is very simple without any counterplay, second one is more creative with tactics.
d4 a5 2. Kd2 Kd7 3. Kd3 h5 4. e4 Kc6 5.c4 f6 6. b3 g5 7. a3 e6 8. b4 axb4 9. axb4 Kd7 10. c5 Kc6 11. Kc4 Kd7 12. b5 g4 13. d5 exd5+ 14. exd5 Kd8 15. Kd4 b6 16. d6 bxc5+ 17. Kxc5 cxd6+ 18. Kxd6 f5 19. Ke5 h4 20. Kxf5 h3 21. g3 Kc7 22. Kxg4 Kb6 etc.. Just created a majority on the Q side and eventually a passed pawn, while leaving three vs three on the king side. Only possible counterplay for black would have been to bring the three kingside black pawns to f4,g4,h4 with allows them to force the creation of a passed pawn. But it's easy to keep an eye on that resource and prevent it if it might become a threat.
e4 a5 2. Ke2 Kd7 3. Ke3 g5 4. d4 f6 5. f4 b5 6. h4 gxh4 7. Kf3 h5 8. c3 Kd6 9. f5 c5 10. b3 b4 11. cxb4 axb4 12. Ke3 e6 13. e5+ fxe5 14. dxc5+ Ke7 15. Ke4 h3 16. gxh3 Kf6 17. c6 exf5+ 18. Kd5 Ke7 19. Kxe5 h4 20. Kxf5 Kd6 21. Kg4 Kxc6 22. Kxh4 Kd7 23. Kg5 Ke7 24. Kf5 Kd6 25. h4 Ke7 26. Ke5 Kf7 27. Kd5 Ke7 28. Kc5 etc..
Gave the h pawn weakening black structure and staying with a majority on the a-f columns while black stays with 2 h pawns vs white g pawn on the king side. Eventually ended up the same way as first game, creating a passed pawn and grabbing black pawns thanks to it, after the tactics arising after 13.e5+. But 6.h4 was kinda just for fun and to show that being a pawn up allows a lot of things. 6.g4 would have been simpler, like in the first game.
A more practical form of your question relates to the Exchange Lopez and Tartakover Caro-Kann.
Euwe did a complete analysis of how White wins the Exchange Lopez pawn ending if you set up the pawn structure with no pieces on the board:
4k3/1pp2ppp/p1p5/8/4P3/8/PPP2PPP/4K3 w - - 0 1
While the Tartakover Caro=Kann is the same thing on the other wing.
In this impractical 8-vs.-7 thing you envision, the side with the extra pawn would surely begin with the unopposed pawn two squares, as the first step toward creating a passed pawn.