Question on Exit Node

I’m a university student. A lot of stuff I need to access is only available via campus internet. My university does provide it’s own cisco vpn, but the client is incredibly annoying to work with. I need to log in and 2fa authenticate every single time I want to use it. So I basically want to setup my own “VPN” by connecting to a computer that is connected to campus internet 24/7.

My current tailscale setup is as follows:
I have a computer that is connected to campus internet and if I’m physically using that computer, I can access various resources like scientific journals etc. I have that computer set-up as an exit node. When I’m not on campus internet, I route my devices with that computer as the exit node. My understanding of exit nodes is that I would effectively be searching with the exit node computer as a proxy, and access all the webpages the exit node computer would be able to access. However, that is not the case, and the websites still show that I am not connected to campus internet.

Is this the expected behavior and am I just understanding exit nodes incorrectly? Or did I not set it up correctly? If this isn’t possible via tailscale exit nodes, can someone recommend anything that will accomplish what I hope to do?

My public IP and ISP changes when using the exit node. I can’t verify the new public IP is the public IP of my exit node device though.