Well, for my project I did partner with someone to make the PCB, as I have about 0 electronics skills (maybe someday...). People like INL or Catskull are usually one-man operations, and they design their own PCB from scratch, so we are still in the "homebrew" spirit ;). But I get your feeling of wanting to build your own PCB on your own - I think you may find some schematics / documentation online to get you started if you want.
During the 80's-90's, AFAIK, Nintendo was the only company authorized to manufacture cartridges. So any publisher / developer would have to buy hardware from them. And you wouldn't be able to make a small lot, but you had to buy tens thousands or maybe hundreds thousands of units - this was a large scale business!
From what I've read, the process was indeed: build the final rom, have it pass a thorough QA session at Nintendo, pay for the cartridges and Nintendo will build them for you.
For the NES era, several companies did manage to make their own mappers. For example Konami designed a mapper to add extra audio channels in a NES cart. But, for some reasons, Nintendo didn't allow third party mappers in US and European releases. That's why only Japanese version of games like Castlevania III have such mappers "not made by Nintendo".
And of course, many unlicensed companies did create their own cartridges from the ground up, like Codemaster and Color Dreams/Wisdom Tree. They were not big companies at the time. So I think even middle sized ones could create their own cartridges and mappers if they had the skill inhouse!