If you haven't, run through this telnet tutorial,
https://docs.microsoft.com/en-us/exchange/mail-flow/test-smtp-with-telnet?view=exchserver-2019, as it
should send an actual email rather than just testing the connection to the connector.
Test with sending mail to 'inside' (domain/O365 tenant mailboxes) and 'outside' (like Gmail etc; we've had/have problems sending mail to Hotmail through a O365 connector/SMTP relay).
Started reading this and noticed it says there is a tool that you can try -
https://testconnectivity.microsoft.com/? so thought I would give it a try and used the Outbound SMTP Test. A few things passed the test but it failed on Real Time Black Hole List? Here is what it says:
Performing Real-Time Black Hole List (RBL) Test
The RBL test failed. Your IP address was found on one or more block lists.
Additional Details
Elapsed Time: 12509 ms.
Test Steps
Checking Block List "SpamHaus Block List (SBL)"
The address isn't on the block list.
Additional Details
Checking Block List "SpamHaus Exploits Block List (XBL)"
The address isn't on the block list.
Additional Details
Checking Block List "SpamHaus Policy Block List (PBL)"
The test passed with some warnings encountered. Please expand the additional details.
Additional Details
An unexpected status code was received from RBL: 11
Elapsed Time: 31 ms.
Checking Block List "SpamCop Block List"
The address isn't on the block list.
Additional Details
Checking Block List "NJABL.ORG Block List"
The address isn't on the block list.
Additional Details
Checking Block List "SORBS Block List"
The IP address was found on block list.
Additional Details
The IP address x.x.x.146 was found on the block list.
Status code: 10
Elapsed Time: 1223 ms.
Checking Block List "MSRBL Combined Block List"
The address isn't on the block list.
Additional Details
Checking Block List "UCEPROTECT Level 1 Block List"
The address isn't on the block list.
Additional Details
Might that be the problem?