This really doesn't contradict the problem of needing to sort out M sites x N devices, where M can be very large.
Whether you do it eventually or do it straight away. Unless you can predict which devices you will have and which sites you will need access to at any given point, then it degrades to needing everything authenticated just in case.
I'm pretty much never too far from either my phone and my security key, seeing as how at least my phone is my car key and my wallet the majority of the time and a security key lives in my backpack.
Sure, M devices can be quite large, but the odds of me being at only one device and not any of my portable devices is extremely small. As long as I have at least one other device I've previously logged in to somewhat handy, I can still easily get in. Maybe that initial login is marginally more complicated, but IMO the ease of future authentications more than makes up for the small bit of initial friction the first time.
And in the rare instance where I'm suddenly on the moon and realize I left practically every other computing device and physical authenticator on another planet, I guess I just won't have access to a DnD tool. Oh well.
Whether you do it eventually or do it straight away. Unless you can predict which devices you will have and which sites you will need access to at any given point, then it degrades to needing everything authenticated just in case.