Certificate won't renew (I think due to the server forcing views to https with outdated cert)

Probably one paired with Phusion Passenger in this case. You should review Passenger docs or their support forums. You could make a worse mess trying things blindly

You might get a clue how nginx was started with below but you really should sort it out properly

sudo ps -eF | grep nginx

I can believe that (though I don't really understand what it means). I know Phusion Passenger is used. The person that was helping do all this for me for years quit awhile back and my attempts to find a replacement haven't worked out so I am trying to keep things going myself for now... (and obviously am a bit over my head).

I figure there is a good chance a restart of nginx may well work (and the reason the "nginx -T" fails is the odd setup just doesn't work for the testing option for some reason) but I am worried it will fail and leave me in worse shape, with nginx stopped and unable to restart (as now at least the sites are still working - though without the updated Lets Encrypt certs).

Copying nginx.conf to the location "nginx -T" wants. Now it fails when it hits a line in the nginx.conf related to Passenger :-/ (which I can maybe understand is could be related to some limitation in "test" versus what would happen on a full stop and start of nginx.

Probably fails because nginx is started using an alternate config file like with the -c option. You would see this in the ps -eF command I described. You probably need to combine -c with -t/-T so maybe like

sudo nginx -c /opt/nginx/conf/nginx.conf -t 

But, as I noted before I am not a Passenger expert so you really should work through their experts to manage your nginx properly

Thanks for all your help. It does seem the direct Lets Encrypt issue seems to be dealt with. Now I just have to deal with a related issue.

There is no way to have the server recognize those new LetsEncrypt certificates without reloading/restarting nginx right?

There is not. nginx loads the certs at startup as part of its config loading. You must reload at minimum to refresh the config.