jugglingphil
Perhaps you have overlooked Charlies advice, "I suspect that OP's problem might be default gateway configuration and/or proxy configuration on client machines."
Charlie is pretty good at his analysis & hits the mark in most cases.
You demonstrated that your sme server does have https access to the net with the command
lynx -dump
https://www.contribs.org/ie "If this command on the server shows content, then https is not being blocked outbound from the server".
So the problem is from workstation to sme or at the workstation, I don't see that rebuilding sme will fix that incorrect configuration.
On sme do
ifconfig
On workstation do
ipconfig /all
and post both outputs back here for comment/analysis.