> Do you have any evidence that this was the reason for the claims?

I did a search on youtube to find these sources ("fauci 60 minutes americans dont need to wear masks") I don't vouch for the channels, these are just the first place I found the relevant clips.

original statement (pretty hard to find):

https://www.youtube.com/watch?v=kQ5oCxP6TUc

his explanation of that statement:

https://www.youtube.com/watch?v=_2MmX2U2V3c

Here's a timeline summary:

https://www.forbes.com/sites/jackbrewster/2020/10/20/is-trum...