If iterate means to repeat, isn't reiterate redundant?

M-W suggests that 'reiterate' means "to state or do over again or repeatedly sometimes with wearying effect"

Maybe it's similar to the difference between 'again' and 'again and again'.

The Ancient Romans didn't think so. Latin had a verb for each.And there's the clue.Their two verbs are the roots of ours .Their 'iterare' was to do or say something a second time.It's from 'iterum', 'a second time'.So an 'iteratus' was a soldier who had been discharged but who was called back to serve a second time.'Reiterare' was to repeat more than just the once.

Iterate originally meant no more than to (do or) say something once again.Reiterate meant to say (or do) it repeatedly, say or do it thrice or more. In practice reiterate came to be used for any repetition and only pedants like 'the Hungarian Sheepdog' make the nice distinction: iterate does serve a purpose in showing that the speaker is saying something for the second time only, not repeating something again and again , as at every opportunity. What politicians say is true reiteration,particularly if they like it or it avoids their answering a question or is sufficiently empty of meaning as to be useful as a slogan or catchphrase ('war on terror' anyone? ).

PS Barack Obama has just said that Senator Clinton's people have now 'iterated it three times'. Now there's a man with precise English !He could have said 'thrice', but not everyone is English