My guess is that it is not a problem with on-prem, it is a problem with MFA. Personally, I have never set up SSMS or ADS to work with MFA on a SQL instance, and I didn't even know you could. I can't imagine why you'd want MFA set up for your SQL instances though as the security can be handled outside of the SQL authentication chain. What I mean by that is that your SQL instances are NOT world facing and as such require either physical access to the building OR VPN access to your network. Since you have that layer protected (MFA for VPN and logging into the physical machines), do you need MFA for the database layer?
On top of that, how do the other applications connect to the SQL instance if MFA is required? Do the applications handle that?
Now, if that is a typo and MFA is NOT required to log into the SQL instances, I would review the logs. Look at both the SQL logs and the Windows/Linux server logs to see what they say happened around that time. It could be something (firewall, web filter, threat detection, etc) saw the connection as suspicious and shut it down. These likely won't be captured in the SQL or server logs, but doesn't hurt to look.
I'd also try connecting with a different tool such as SSMS just to confirm if it is a tool problem or something else. Another thought is it MAY be that ADS doesn't support the version of SQL Server you are connecting to. Offhand, I am not sure what versions of SQL Server it support, and I am unable to find any reliable documentation on that.
And my last question - you ARE connecting to a Microsoft SQL Server instance, correct? Possibly a dumb question, but if you are connecting to Oracle or PostgreSQL or MySQL, ADS is NOT the right tool for you.
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!
I recommend you NEVER run "random code" you found online on any system you care about UNLESS you understand and can verify the code OR you don't care if the code trashes your system.