The way to grokk the Red setting is that it's "New York in the 1970s" but taken to extremes (as usual). As famously said in Brooklyn Nine Nine, "New York in the Seventies was basically The Purge." New York was rich in the 1950s, it started getting rich again in the 1980s, but inbetween things got almost unbelievably dystopian and poor in places (the city came within a millimetre of going bankrupt in 1975), even though the reputation for being a cool city never entirely went away.
So Night City gets micronuked, there's fighting around the city in the Fourth Corporate War, and the city resasserts its independence despite pressure from the NUSA. A large chunk of the city's populace vacates (as much as the nuke "wasn't that bad," it's not Fallout, it's still kinda off-putting to live right in the place where it went off, unless you're so poor you have no choice), Arasaka gets booted out - to be honest, a relief given how much of the metaplot they end up driving to the point it gets a bit silly (as great as the video game is, I could go a long time without ever hearing Keanue Reeves' pronunciation of Arasaka ever again) - and the city is trying to hold on and rebuild. But it can't due to a lack of investment, megacorps losing faith in the place etc.
That gives you the Time of the Red: the city is a bit more Wild West, it's reliant on the help and alliance of the Nomads to keep going (which obviously gets forgotten about the second it's inconvenient), megacorp influence is there but waning, but also some of the lesser megacorps use the time to build up some serious influence in the city (so you can move away from everything being about Arasaka the whole time), and Militech becomes the new Big Bad, as they are angling towards working with the NUSA government to annex Night City. The vibe is Dogtown from Phantom Liberty but applied to the whole city (more or less).
But you can see the corporate influence seeping back in, the rich stayed away for a while but have come back via Pacifica (and it's amusing knowing how that turns out), people are starting to realise the city could be on the upswing and want to start building again or hoovering up land in the cheap to sell for hundreds of times that worth later on, even in the nuked-out city core (that's what the starter adventure in the Jumpstart Kit is about), and music fans etc are still visiting the city because, no matter what, it's still the city of Samurai and Johnny Silverhand and it still has a good music scene. It's also a handy place to do back-room deals, people from the NUSA and the independent States and Mexico can meet up there without being closely monitored (think of a grubby Singapore). It has a lot of potential which we will see realised in 2077, when it's surpassed its heights in the 2020s, despite having horrendous crime.
I think looking into the 2077 timeline and seeing that Night City does eventually return to the glory days (well, kinda) is a good thing to do before launching a Red campaign (which may or may not align with the official timeline, of course). The city cuts a deal with Arasaka to prevent Militech and the NUSA annexing it in return for Arasaka coming back and having maybe more influence than they ever did before, which is something the players could even get involved with. If anything, the Red timeline might be a bit too early for that, but you could probably move the Red setting to say 2065 without it being too different, maybe rebuilding is a bit more advanced and that's it.
