If you want a detailed explanation of the Amazon show, I did it in this post here but the short answer is it's actually good and not really woke at all.
This post is more discussing a something that looks to be the future of the franchise in terms of lore so there will be spoilers for the show.
Basically in the show, ALL lore related to Fallout New Vegas has been erased. The fractions are either not mentioned at all or severely diminished in terms of the NCR, some things in 1 and 2 have been changed (like Vault 15 which Shady Sands was built around is not mentioned at all) and it looks like they are only using the setting of New Vegas in a season 2 as they show the strip at the end of the season.
This is one thing, but I also recently heard of a Fallout 4 update and DLC planned where the update is optimising for 'next gen' consoles and the free DLC will introduce a new fraction to the commonwealth to stop...
..the fucking Enclave...
They seem to be only keeping lore cannon completely in Fallout 3 and 4 so does the Legion exist any more? Are the Followers of the Apocalypse still around or they gone too? Does any of the characters we liked from New Vegas exist either like Boone, Veronica or Cass?
The other thing that urks me is Fallout New Vegas was the best written and world built game in the franchise in competition with 1 and 2. The fact they are bringing back the Enclave for 4 just makes me think the writing staff at Bethesda aren't that creative at all and they're just rehashing old plots again and again than try some other direction or a new take on an old fraction. It's REALLY not giving me hope for Elder Scrolls 6...
TDLR: Bethesda may be doing what Disney did to Star Wars Extended Universe and erasing lore not made directly with Bethesda
Is it?
Or is it just that in western franchises quality seems to decline without it?
I'd argue it's pretty important.
You can have good stories with bad continuity, but having good continuity generally means the quality is likely to be higher in general
I wonder if you could chart it out, a continuity vs care graph? The higher the care the higher the continuity.
I think a lot can be forgiven until you hit character rewrites (turning them gay, making them dumb, or even super smart.)