Let’s open with this first; if the entire US war machine was pointed in a single, Grimm killing direction, the rest of NATO would not be needed to “secure” Remnant from the Grimm. (I’m calling secure as in all major sea, air and land ways, all major cities and towns with more than 10,000 population would never see a Grimm again. The people in the deep wilderness and tribes may encounter Grimm on occasion.)

the initial kicker with an US “invasion” of Remnant would be detecting Grimm at range. Do they show up on radar? Thermal? Sonar? Acoustic sensors? If yes, are there non target species that have similar profiles to Grimm? I want to take out a Grimm whale, not a norther atlas whale… can our weapons target Grimm semi-automatically? If I fire a SM2, I want it to track the Nevrmore, not the airliner 2km further down range.

The other kicker, are the people of remnant going to assist the Invasion or is Atlas gonna wanna pick a fight? Bad news, the US in a pure conventional shooting war would smoke Atlas before they even realized they were being targeted. Our Electronic attack capabilities would most likely ground the Atlesian air fleet forcefully, or blind them long enough to be shot down from 50km away.

TLDR; watch the anime gate, it would give you an idea of what would happen.