With the knowledge that the new show was to be in an alternate universe, people keep saying this was the only way it could go, which while I don't think the show's concept is horrible, I think there were other ways possibilities and I keep thinking about these two so I just wanted to put them out here to see what people think. Premise 1 is the option I personally like better, but premise 2 is the sort of thing I think people like Amazon would be more likely to go for. I want to make it clear that I don't think I'm a better writer than Martin Gero or anything, I've just been thinking about these concepts and I want to see what people think
Premise 1: It has been 15 years since we last heard from our heroes, in this time, the Stargate program has been made public, and all the old threats have been dealt with. While Earth's fleet of ships patrol the galaxy, offering aid, and dealing with the remnants of old enemies, a new team of explorers has been assembled. The Stargate was made public some time ago, and the new Stargate Program consists of the combined efforts of Earth's major powers. It remains largely American-led, but like the earlier Atlantis expedition, the best and brightest have been recruited from around the globe.
The show follows a new team of explorers, going on missions through the gate, this show would be very episodic like early seasons of SG1 where the frontline team explores a new world each episode. There would be plenty of opportunity for some returning faces, for example I know a lot of people have suggested General Samantha Carter as the leader of the SGC, which would be a good way of incorporating a legacy character without having them steal the show. Additionally, you could bring in old characters for individual episodes eg: there's a really difficult language that the team's archeologist can't decipher, so Professor Daniel Jackson is brought in to help out.
The show starts out relatively lower stakes, where the team is just exploring individual worlds, but as the season progresses, they start to unravel a conspiracy spread out across many worlds that no one else saw because they were too busy looking at the big picture... (this could either be a resurgence of an old enemy, eg: the Goa'uld have set up spies all over the place trying to subtly outmaneuver the humans, or it could be something completely new.)
The main issue with this idea is in creating stakes when you know that there are powerful ships flying around out there. But space is big, the advantage of using the gates is how quickly you can get around, so you could easily use the old "the nearest ship is three days away at max speed." But you can only do that so many times. This would at least provide an opportunity for some epic moments eg: the team have defended to their last and the enemy is about to kill them, when suddenly the Odyssey arrives and strikes them from orbit
Premise 2: We start off by establishing the formidable fleet that Earth has. When suddenly, a wave goes out across the galaxy and all of the ships hyperdrives are rendered inoperable... (ala discovery's "the burn" except this time we don't make it caused by a crying child, and there is a basis in Stargate for this sort of thing given the Attero Device.)
Now all of the ships are stranded across the galaxy with no way home, Earth is practically defenseless, the only way out of this crisis: the Stargate. Funding was pulled over a decade ago and the whole program was mothballed. But now, they need to pull the gate out from the warehouse where it has been gathering dust and get it operational again. This effort is spearheaded by one of the old cast such as Carter or O'Neill, and some technicians eg: Walter Harriman and Sgt. Siler come back to help get it running again, but the main cast is a whole new team who's goal is to use the gate system to weed out the origin of this mysterious attack and put a stop to it...
So yeah, I personally think there is plenty of opportunity for a new show to be in the same continuity as the old shows without it being all about the ships and how powerful Earth is, the galaxy is a big place and there are still countless unexplored worlds out there...