NYT wants to block them because they don't want to be held accountable for spewing Trumps propaganda. If you are truly locking it down, just only make your news accessible via login.
The NYT already has a very strict paywall, so there's not really a point in blocking the Internet Archive. Things are very different for The Guardian though, since it has absolutely no paywall on the web.
NYT's paywall is softer than FT's or WSJ's - just deleting domain cookies is enough to bypass.