Is there any way to see if a Pending New Member message or a Welcome message has actually been sent to members asking to join a group?
Maybe.
If the Pending or Welcome message is listed in the (pending) member's Email Delivery History as the most recent Successful or Unsuccessful then you know its fate. But a subsequent message could supersede it there,
Too, I discovered recently, it can take a while (between 1 and 24 hours, in the case I noticed) for the Pending notice to show up as Most Recent Successful Delivery. I don't know if that's just an updating issue in Groups.io or if it relates to some kind of time-out for delayed rejection by an email service.