A Reboot?
The driving plot of Saints 4 was, besides killing Zinyak, trying to find a way to restore Earth from complete destruction. Keith David (as himself, natch) believed that by betraying the Saints, Zinyak would be able to bring the planet back as it was before the invasion, and at the end, it was revealed to be all a lie, and there’s no way to bring Earth back. The Saints are, essentially, the last humans in existence (that we know of).
That may sound bleak, but Saints Row is also a series known for pulling last minute swerves. They spent two years letting people think Johnny Gat had died in Saints 3, only to bring him back for the fourth game, so it stands to reason that they could come up with a way to bring back the Earth here and act like Saints Row 4 never happened. Going off of the famous people he’s kept in stasis, Zinyak had the ability to travel through time.
It wouldn’t be entirely impossible for the Saints to figure out how to work it and make it so that they save the world in time to beat the Zin (and also take Zinyak’s head), in the vein of X-Men: Days of Future Past. The President’s approval ratings were already down before the Zin invaded, so an alien invasion can’t have really made them all that better, even if the invasion was repelled. This could end up being a soft reboot, of sorts, for the series.
Something would have to happen at that point to where the Saints end up deciding to go into the private security, world saving business. No one else appears to be leading MAYHEM besides Brimstone, and it’s possible that she was able to either buy off the other Saints or just straight up kill them to take all the power for herself, explaining the similarities with the logo.