What do you think would happen, if only one team had minions? Which team would win?
On the one hand the team with minions (let's call it Team A) would have the lanes always pushed and they can attack towers easily. However, the other team (Team B) has access to way more farm. Team A can only farm in the jungle, and get xp by champion kills. However, Team B has no way to push towers until late game, when tanks are tanky enough.
I guess, the meta would be completely different:
Team A would choose to go 5-man mid, to exploit their pushing potential. Or go to lanes, and deny the opponent farm, with very long range abilities.
Team B would surely have at least one champion per lane, to farm and prevent the towers' destruction.
The champions would also different. There would be hardcore armor tanks on Team B to push towers, and hard tower pushers (i.e. no crit, only ad and as).
Team A would have champions with extreme range, to poke the opponents and destroy towers.
I can really not tell which side would win, but I'd go for Team A.
I think that, if such a mode existed, it would be really fun, and require good strategy, if of course huge balancing were to take place, cause otherwise only one Team would win each time.