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Oh, for sure. I guess I assumed Bob made the introduction by email. So OP could reply to that intro email with a “Thanks, Bob! Alice, pleasure to meet you, I’ll send you a separate email to introduce myself and blah de blah”.
Acknowledge the email and thank Bob, then drop him from the convo. Regardless, Bob doesn’t need to see the sausage being made.
Hindsight is 20:20. Regardless, I think Bob is overreacting, assuming your email to Alice was in the range of what I personally would write and nothing you said was sparking red flags to me l.