Android TV can run Tailscale or Wireguard natively. AppleTV has a native Tailscale app, and I think you can also use Passeportout for Wireguard on AppleTV but I haven't used it. Alternatively if you're on the go a lot and want to use a streaming stick in your hotel you can use a travel router that supports VPNs like GL.inet.

Airplay and Chromecast are a different story. Maybe someone else here knows different, but while it's not literally impossible it doesn't really work because of mDNS. A layer2 VPN might, but not so much on Tailscale/Wireguard.