I would love a Gorgon for taking on Corpus, but as you said, it's a random reward.
Can't you buy the blueprint with credits? I've done that with almost all of my weapons.
Also, count me in with preferring weapons dedicated to certain factions or modes. I prefer having full loadouts for each faction rather than using A/B/C sub-loadouts, which can easily be forgotten since there's no UI support for them.
But my dual vastos and glaive are not only useful for a variety of situations, they're also best in class for most situations. At least for my arsenal, so I do have separate builds for those.
My current setup:
Volt: My main frame, with the most utility across the board. Also my dedicated Corpus hunter. Chain lightning kills or at the very least stuns, the room-clearer is powerful as hell, and speed is always appreciated. Shield also comes in handy on occasion. With my volt I rock a magnetic Gorgon, magnetic+fire Kunai, and a magnetic Glaive.
Saryn: My go-to for hunting Grineer. She's setup with a Poisonous Latron, Dual Vastos and a fire Glaive.
Ember: My go-to for infestation missions. She's got a Dread as primary, with the Dual Vastos and fire Kama to round things out.
AT this point I'm just leveling everybody up, polarizing if needed, and looking for weapons that can round out these builds a little better. Like something semi or full auto to replace the Ember's Dread for laggy matches - thinking Soma here. And maybe something besides the glaive for my Volt, but I do like tossing the thing to take down cameras.