Except there are cases where such a thing does not exist, or that they are not identical - I'll note the Underbarrel weight here - the accessory version applies only to full bursts, whereas the modification version applies to all recoil. And that doesn't actually answer the question, because there's no rules text that says to do so!
Yes there is, you're choosing to ignore it. If they acted differently such exception would be noted in the Mods vs Accessories section in Arsenal.
Not quite. There's nothing that states that the modification rules follow the same rules as accessories - there is something that spells out a difference between mods and accessories of the same name. This is not the only difference - for example, in some cases, the two function quite differently, as with Underbarrel Weight. Nor do the rules tell you to look at an accessory bearing the same name to determine if it requires mod slots. You choose to run it that way, fine - but it's not written that way, and given the application of a different limiter, it is likely not intended that way either. That said, there's a thread about this in the rules forum, started it a while back because someone wasn't sure.
As for the bayonet mount, you're misreading it. The modification does not give you the bayonet, and is not the mod version of an accessory. The modification gives you a mounting slot - and would actually be quite pointless if it required a mounting slot.
Also, without mentioning any variance, the gas vent mod indicates how it interacts with certain barrel-mounted accessories, which it would not if the mod took up the barrel slot and thus made the interaction impossible.
Perhaps it's best to continue the discussion there if you must, rather than continue to derail this thread further.
My contribution:
Ares Alpha w/
-- High Velocity upgrade, 2 slots
-- Gas Vent 2, 1 slot
-- Personalized Grip, 1 slot
-- Additional Clip, 2 slots
Put it all on a gyro stabilizer, kicking you up to a good 11 compensation, and suddenly you're looking at 17DV full bursts without recoil. Load your clips with whatever ammo you like - perhaps lethal and non-lethal, or flechette and APDS, or AV and Ex-Ex... So many options, there. If one type of ammo is plenty for you, swap the extra clip out to converting it to a drum load, since this thing is gonna throw a lot of bullets downrange.