Okay, long post...
A probably somewhat reliable (but slow) solution
Open the Netflix App, wait 10-20 seconds to see if it loads. If not, close the app. Open any app other than Netflix. Close that app. Open Netflix again. Repeat maybe a dozen times, until it works.
What's the problem? (mostly guesswork here)
The Netflix app is really two systems: the "client" on your actual box communicating with Netflix's server. Since the problem seems to have appeared on all kinds of user-side machines without being tracable to a particular update, it is very likely that the change happened on Netflix's end. The problem also only appears while starting the app, so it's likely related to the login process. At least for me, the problem appeared around the time Netflix announced that they would crack down on multi-household accounts. It's possible that they are doing that by adding another step to the logon protocol, one that is only sometimes required. If that's the case and the Netflix app on your box does not support that step correctly yet, it will simply fail and continue to "load". But again, this is just a best guess.
Why do some solutions seem to work?
IF this explanation is correct, it's really just a matter of repeating the process until it works. But for that, the Netflix app needs to be completely closed after every attempt. By default, it seems to remain active in the background and thus you are not actually starting a new login attempt every time you open it. Solutions like software updates or system reboots will obviously shut the app down. It's much easier, though, to just briefly open another app because it seems that only one app can remain active in the background at a time.
What can we do long-term?
Not much, really. Netflix and Sky (or whoever is developing the app itself) need to sort this out and probably roll out a patch. For that, they need to actually be aware it is happening.
I hope this is helpful to anyone. No guarantees about anything.
