For one, alien species deal with the SGC as if it represents the entirety of Earth, when for various reasons it doesn't. Oh, sure, there's the IOA, but that organization was clearly a patch job to lend airs of planetary legitimacy to SG-1 and the SGA's activities. They were, more often than not a source of episodic or mini-arc/recurring conflicts, much like the Senate and NID were to supposed to take the weight of interplanetary military and exploartion off of the USAF's shoulders as such was not their intended remit earlier on in the series. And we saw where those led (Kinsey and the Trust rather thoroughly ruined both).
There was no massive true behind the scenes unification of Earth's interests, or even public reveal to lead to some new global governing entity (Because somehow that for a single planet is beyond the scope for a series that ends up dealing with Pan-galactic upsets to status quo and entire collapses of various multi-centennial/millennial societies/governments). Either Apophis' or Anubis' violation of Terran Space could have been the impetus for a great public reveal, nevermind the Wraith Superhive Ship. That's just not what the series was truly about.
Honestly, it would have been better if the SGC had a lesser overall impact, galactically. They were not set up to be a massive interstellar infantry, and even the Tok'Ra were better suited for Geurilla and long term tactical warfare than the Tau'ri. Push the advent of the F-302 back another season and a half, and let the Prometheus be the ultimate product of Earth's innovation and engineering, with the initial BC-304 being revealed at the end of Atlantis' run. Don't overcome the entirety of the System Lord's, and instead be a victory over a faction of the Goa'ould while creating the seeds for a greater rebellion (free Jaffa and greater latitude/opportunities for the Tok'ra, along with uniting other galactic powers that were otherwise separate such as the Orbanians, the Tollan and the Hebredian to name a few). Sure you might find a Humans-Hell-Yeah story more engaging, but it could have been written so humanity and the SGC were the glue that stuck them together.
As a result, the plucky group of Stargate/Homeworld Command quickly (far more so than should have been possible) overturned all the Apple carts which necessitated plot devices or cloning and scaling various threats. Shatter the unity/effectiveness of the System Lord's, Anubis crawls out of his hole. Break the control of the snakes over the galaxy, the hydra that is the Lucien Alliance comes into play. Defeat the primary distraction of the Asgard? Well now we have to keep the Roswell Grays out of the picture as we wrote them in to be too overpowering otherwise. Oh, and we'lljust reintroduce more advanced versions of those replicators later in the Pegasus galaxy (*thumbs up Lanteans* Good job!). It honestly seemed to begin to suffer from "Next time on DBZ..."-itis.
To fix the problems with the SG series, its growth would have had to be capped, and quite frankly it would have needed to be more like Babylon 5 rather than Stargate. Frankly, I would much prefer to pick up my sack of salt, employ the use of a construction crane to hold my disbelief up and just enjoy the fun that was MGM's/Sci-fi Channel's last real pride and joy.
That said, having other world's mix with Stargate for too long, or mesh too much, really ends up distracting from the story that could otherwise be enjoyed with it in moderation or nearly entirely ignored. So, occasional omakes with the SGC wouldn't be too distracting, one or two bits of SG tech or lore ending up in the main story could still work, but anything else I suspect would be more likely to sour the pot without a serious retooling of both ends of the plot thread. Which is a shame.
