I am on an older Mac Mini with macOS 10.13.6 with Tailscale 1.36.2 from the Mac App Store.
On the App Store it always says “No updates available”. If I search for Tailscale, I can find there is an Update, but when hitting Update, it just spins for hours.
I even remotely got into the Mac Mini with a different VPN service and quit Tailscale, but retrying the update still fails.
I deleted Tailscale, and was able to install 1.38.2, but I fear this will keep happening in the future.
I ended up using Screen Sharing (via Tailscale + VNC in my case because the machine is remote) and dragged an updated version over to the 10.13.6 machine. The tricky part is that as soon as you quit Tailscale you’ll lose the screen sharing connection. Hence a VNC to access the machine outside of Tailscale and do the update. ARD is way expensive so I use Edovia’s Screens 4 which is a bit quirky but does the trick once you get used to it.
I’m not sure what you mean here but you were obviously able to download the updated version so the above approach might help.