...but folks in the Delta Q. aren't going to do trade negotiations with a Borg nor with strangers in uniforms. They will do it with a native to the DQ like a Talaxian. Seven could provide info on each species but It was Neelix who'd they'd sent out to negotiate trade. It was the exact reason Janeway called him Ambassador and put him in charge of ships inventory. Who would suspect Voyager was hostile if they send a kind but a dimwit like Neelix to represent them? People would do trade with him cause he's a nonthreatening native. Trade thru reverse psychology.
Yes, being a cook for nearly 150 or so would keep him VERY busy. It's another reason why supply runs and inventory would be part of his job as well.
From Season 4 on, how do aliens know Talaxians are Delta Quadrant natives beyond taking their word for it? And what do they care about a race that is from 10,000-30,000ly away "but in their quadrant" vs. a race that is 30,000-65,000ly away "but from another quadrant"? His use there is just as someone experienced as a trader, salvager. Where he's from is no difference to them since he's no longer a local this side of the Nekrit Expanse.
I do agree, Neelix was kept busy as the ship's cook and quartermaster. It doesn't get recognised as such. He often mentions coming back with a bunch of food and other provisions for the crew.