That's interesting. Seems like it should work. I wonder if there is a networking problem, like the SMTP port being blocked somewhere between you and Sendgrid.
I would suggest that you try making a direct connection to Sendgrid's server with a command line tool. This is pretty easy to do and it will show you if something is blocking your connection.
Assuming you are on Windows, go to this site and download puttytel.exe.
http://www.chiark.greenend.org.uk/~sgtatham/putty/download.html
Open the program and change the highlighted settings. This should open a connection to Sendgrid in exactly the same way as your NVR would do.
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When you click Open, you will hopefully see a new window open. The text in this window is the "I'm listening" message from Sendgrid's email server.
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If you see something like that, good news... You may have a configuration or firmware problem, but there's nothing fundamentally blocking access to Sendgrid.
If you can't get this connection to work, or if there is some kind of error message, then something is blocking port 587. (Firewall, etc.) Since you have tried every setting combination it seems like a good idea to verify that something like that isn't going on.
Good luck!