Also, try to connect to one of the servers using its IP instead of its name. You can find a list of servers by clicking on the red circle. Ping the server to retrieve its IP:
ping server_name
Some servers disables their ICMP ping and can result in a false negative.
Here's a list that can be pinged:
electrum.jochen-hoenicke.de
electrum.blockstream.info
b.ooze.cc
bitcoins.sk
electrum.eff.ro
OP, if you're able to access the website, then the update checking should not fail as it is hosted on the same domain. There might be something specifically blocking your Electrum from accessing the internet instead of a filter on your network.
Go to Tools>Preference and check "Write Logs to File", restart Electrum and try connecting to a server again. Next, go to the data directory (%appdata%/Electrum) or whichever directory you're storing your files. You will see a folder called Logs, open it and open the latest file with a text editor. You can scroll to the bottom and look for any anomalies related to the connection attempt.
Alternatively, you can try using Wasabi wallet. It isn't as easy to use as Electrum IMO but it is definitely better in terms of privacy in the first place.