I don't use Nord personally, but I understand they are introducing Wireguard as a protocol so that subscription might not be going to waste:
https://nordvpn.com/blog/nordlynx-protocol-wireguard/
I think getting it up and running as a network wide client might require some Linux skills right now, but it looks like they are working on producing material and instructions for other OS and routers.
More generally I suggest you think about something called policy based routing on whatever network wide solution you decide upon. you say you'll "stumble upon censored content, I switch" - that's going to be a PITA to do ad-hoc every time I think.
Policy based routing on your router of choice will use rules to say automatically where traffic will go, either via VPN interface or without.
So for example, I have a rule setup for one particular machine's internal IP to always go out via a VPN interface because it's purpose is pretty much just downloading and moving media files up to the cloud - nothing that being always behind a VPN will restrict. I then have another rule that says no matter which machine on my network, if the destination is my newsgroup provider then route it via VPN. I kick off some downloads on different machines so this means I don't have to worry about which one I'm using, downloading is behind VPN.
In your use case maybe you can implement similar rules the other way around. So for example maybe the default is always behind the VPN except the IP addresses of your games machines because you want the 'cleanest', fastest connection and you don't do anything you worry about requiring a VPN. then add some rules to say if the destination is iPlayer, hulu, your online bank etc. then it routes without VPN. Taking the time to set that up properly will reap rewards long term.
Again, because I don't know Nord nor Wireguard that well I can't suggest a device in your budget that will do all this.
I think a Pi with PiVPN might be able to do this in budget but not sure. There's I think one particular Asus router that will do this at the speed you want with custom firmware for $200. Draytek, Mikrotik and Ubiquiti might have off the shelf devices that have the capability as would high end enterprise stuff but I don't know enough about them. But the one that I know does work, because I use it, is a $300 mini PC from the likes of Proctelli (other no-name Chinese copies are widely available) with an AES-Ni capable CPU (better OpenVPN performance) running pfSense. Other free firewall distributions will do the same on the same box like opnSense, Untangle, Sophos etc. and netgate sell appropriate pfSense appliances with proper support. Please don't expect all this to work without needing to get familiar with networking although some distributions/devices come with a steeper learning curve than others.