How can I eject a Time Machine drive quickly?
There is a 3rd party app named Mountain that gives you control over all your connected volumes. It's not free, selling for $7 but they have a free trial period so you can test drive it. Mountain creates a menu item that user can use to mount, unmount, eject or load volumes. I can't guarantee it is faster than your method but I use it to mount or unmount stubborn drives.