You are getting the gray start button because you are unable to connect to the lobby server.

So, the support rep was partially right... in a roundabout way. There is a connectivity issue between your client and the server. The big question is, just where that is coming from.

Check the logs for your security to see if it is blocking anything when you are trying to get in game. If you see any XIV executables getting blocked, need to white-list them, giving it internet access and such. Might want to white-list the SE folder in your anti-virus as well, just in case. Some like to run them the first time in a sandbox to test for vulnerabilities before they actually allow them to run normally--this testing may take long enough for the launcher to time-out and crash on you, causing the .exe to close before you are aware your AV is holding it up with the scan.

If you see some IP's getting blocked when you are trying to get in game, google the IP address along with myip.ms (myip.ms 124.150.157.190). This should get you some links that give details about who owns that IP address--if you see SE in there, white-list that IP (or you can do the subnet if you want, provided you have verified they own it).

With any luck, you find one or both of those things happening and get it resolved. It happens to me when my firewall detects an updated .exe and I forget to update the enable/allow list for the new file.

Not sure if it could be related, but others have had various issues with the launcher due to outdated browser technologies. Make sure your add-on stuff like Flash, Java, .NET and such are up to date--even your browser itself. This is more applicable if you are seeing things like scripting errors, but with all the security holes found in these technologies lately, it's a good thing to do in general and may prevent such problems from cropping up once your connectivity issue is resolved.