Want to Resolve Discrepancy between Mail Accounts and Connection Doctor

In Applemail Preferences, all of my email addresses are listed as online. and there are no error icons in the sidebar indicating otherwise. But in Connection Doctor disagrees. The addresses that are not connected change around, but they are always Type: SMTP. IMAP accounts are connected.

I am able to send and receive email, although I get reports from some people (at some times) that they haven't received an email that I know I sent.

While some recipients seem not to be astute technologically so there may be reasons on their end that they're not consistently getting my emails, this discrepancy doesn't make sense and I'd like to rule out any problems on my side of things.

Background: In January my previous email server closed, so I switched to my website host for handling my email server needs (I have 10 addresses associated with two domains). I worked with my website support team to clear up some account info, mailbox behaviors and server settings glitches, but we couldn't get past this. They're as perplexed as I am why accounts look online but Connection Doctor disagrees -- so they recommended that I turn to Apple.

I'd like to know

1) Why this discrepancy might this be happening

2) What can I do to resolve the discrepancy (or guide my website host in what to do)

3) Suggestions I can pass on to recipients to do on their end (over and above checking spam/junk folders which I already do, with mixed results).