The Near East and the Middle East are two different, albeit overlapping areas.

The Near East includes North Africa and Turkey, but not Persia (Iraq, Iran, etc.), while the Middle East doesn’t include North Africa (except Egypt) or Turkey but does include everything else east of the Indian subcontinent (Afghanistan isn’t technically in the Middle East, but usually gets lobbed in anyways).

“Near East” didn’t fall out of use because of “US influence”, it fell out of use because 98% of the time we are talking about e.g. “conflict in the Middle East” we are talking about the geographical region that includes Iraq, not the one that includes Morocco.