Dano,

Would it be possible to have that search option after we select a
member in the member search?

As explained in GMF, the list of postings by that member is only one step away once you've opened the member's subscription page. Or two clicks if you go by way of the member's profile.
https://groups.io/g/GroupManagersForum/message/15620

Alternately, would it be possible to have the option under message
search to search for sender, probably including email address?

I concur with the general desire to have more tools (filters) in message search, including the ability to search on the From address or Display Name of the messages. That would benefit members and not just moderators with access to the members list.

Shal