The G-Man sent Gordon Freeman to fight the Combine in Half-Life 2. He also gave Black Mesa the sample that caused the resonance cascade that got everything started in Half-Life 1. So, it’s not clear what the G-Man’s actual goal is with regards to the Combine.
I haven’t played Alyx, just the other games, but the overall situation sounds like a Cold War to me. It’s not that the faction the G-Man represents wants the Combine to stay on Earth, they just don’t want to directly confront the Combine with their own forces, because a direct confrontation would be too expensive/risky for their other interests. Kind of like how the US wanted the USSR out of Afghanistan in the '80s enough to fund and advise local rebels, but not enough to send in the 7th Cavalry.
You’re probably right. Still the G-Man was able to take Alyx into the future which is an impossible task by any mortal person. If he was able to do that then why couldn’t he get rid of the Combine on Earth? Both things are impossible but it seems like one is more possible than the other.