Of course, it's completely possible that some settings in Mail are hosed, and that it could be easier to setup Thunderbird than to fix Mail.
Part of my problem is that installing Thunderbird is a stopgap - Mail still needs to be fixed (or abandoned altogether). Installing/configuring another mail app probably isn't a necessary "emergency" step - those accounts can probably be access by a web browser-based login, in far fewer steps.
My focus is on fixing the problem. The OP asked how to fix Mail.
As a matter of principle, I'm not big on "one size fits all"/"this one worked for me" suggestions. These often do not work in complex systems, as there are so many potential causes. Maybe the shotgun/try this approach works. If it doesn't, you're back to square one, rather than working your way through a logical troubleshooting tree.
"Cannot send mail" (on any computer or OS) is nearly as common a symptom as fever is for a sick human being - nearly every illness known to medicine seems to cause a fever. "OK, you have a fever. What else? Headache? Stomach ache? Rash? What did you have to eat today..." In this case, we know that the problem began after configuring a new mail account and installing a new router. We know it works correctly for the newly-configured account. Is it reasonable to assume, "Must be an Apple bug," and skip over the other evidence?
Yeah, you could use "install Thunderbird" as a troubleshooting tool. If it doesn't work in both Mail and Thunderbird, we look for something common to both (like the router). If it works in Thunderbird but not Mail, we can focus on fixing mail (back to square one). But it's not the first thing I'd do. I'd run Connection Doctor. I'd check the SMTP Server settings. And I'd look at that new router. I already have symptoms pointing in those directions.
In the meantime, you said, "if the user wants a POP3 connection" - POP3 is for receiving email, not sending. The OP did not mention a problem receiving, only sending. Follow the evidence.
