It’s not really that much of an influence, especially if you consider the amount of English words that went into Japanese.
Exactly.
As a kid, I lived in the Japanese islands for six years. Yeah, it was a long time ago, but as I recall, the Japanese used quite a few English words, and even phrases. It was necessary, as their language didn't have words for many modern day things.
America also exerted a huge amount of cultural influence on Japan in the post-war era.