I don't agree with or believe that. It's too simplistic. The US didn't need Britain's military help. I believe Blair did it because he wanted to, thought it was the right thing to do and was a low political risk. He even
gave a speech about his doctrine of moral interventionism in 1999, way before Iraq. He was of course staggeringly wrong but it was not clear at the time, especially post Serbia, that military action against a country to kick out a dangerous, cruel and corrupt regime, was the morally wrong thing to do (and it really only became obviously the wrong thing to do because the outcome was so incredibly destructive).
(I didn't support Iraq because I couldn't see evidence of why it threatened the UK. All the sexing up of dossiers was designed to manufacture such evidence BTW).