Do you know just how old is that Sony NSX-32GT1 TV, or which version of the Google TV operating system it's running? Just looked up Google TV online and saw that Google killed it off back in 2014. Apparently the Google TV OS is Google's proprietary, branded user interface with various, also dated versions of the Android OS running underneath.
https://en.wikipedia.org/wiki/Google...rating_system)
So yeah, perhaps the out of date software running on that TV is the problem.
Try going into the Settings menu on the TV, go the Accounts menu, and delete any or all the accounts that might be listed there. Then using a computer or mobile device (presumably more modern) log into your Google account Security menu, go Devices, and delete that Sony TV entry (if there, might not be since you couldn't log into it using your account). Now back on the TV try adding your Google user account in the Settings menu again.
Are you sure your TV is properly connected to your home network? Be sure to double-check your router's setup page that your TV is connecting, online access is a given. Sony has online manuals available online for your model, might be helpful in just starting over the initial setup process.
https://www.sony.com/electronics/sup...-32gt1/manuals
If there's a reset button on the casing or reset option in the Settings menu, you might want to use it to just reset the TV back to its original defaults.
A very last resort option, seeing as your model does have a HDMI port, you can use an external dongle like a Chromecast or Roku. Note you don't need to have your TV itself connected to your home network, it's the dongle connecting over WiFi that enables you to view online, streaming content. Your TV is then just getting it's signals from the dongle. Since you don't connect the TV itself to your home network, the TV doesn't use Google account access since all online access is through the dongle. It's important to note the significance of this setup -- you don't use the TV's remote to uses any apps and such running on the TV; you instead use your phone and the apps you have installed on it to interact with the Chromecast or whichever set-top dongle you chose. (i.e. you don't use the Youtube or Netflix or whatever app icon is on your TV's screen, you use the apps on your phone to view your online streaming content.
I've got an Android TV set up this way, with a Chromecast plugged into a HDMI port. The TV has no Google account set up on it any longer, we all just use apps on our mobile devices to interact with the Chromecast. Actually like this situation might work out well for you, the version of apps you use on your phone will be updated/upgraded more frequently and be more current, while the apps on your dated Google TV will more likely than not be older versions. (i.e. the Netflix app on your phone will be more current to so it supports the latest features and interface of Netflix's online service, while the TV version of the Netflix app will be out-of-date and will have limitations when viewing your Netflix content.)