As others have probably said (didn't read the whole thread), backgrounds ARE professions. So yeah you are overlooking the option big time.
Just look at the list of PHB backgrounds, all of them answer the question: "how do you make a living"?
Most of the PHB backgrounds answer the question by giving you a fairly regular job, for which you can assume there is either an employer or direct customers: artisan, entertainer, soldier, sage, acolyte... Other backgrounds are ways to make a living in a more general way e.g. criminal, charlatan, and finally some of them are about simply getting along and provide for yourself e.g. hermit, urchin, outlander.
Now, clearly 5e moved away from 3e approach of having a "one-size-fit-all" skill to cover everything related to a profession. Instead, each background gives you multiple skills/tools/language proficiencies to represent a set of things that practicing your profession makes you good at. That is certainly different, but the "one-size-fit-all" option is still there as a variant rule (see Background Proficiency in the DMG) and I think many DMs actually use that.
As for learning new professions during the campaign, downtime rules allow you to learn tools/language proficiencies and feats allow you also to learn skill proficiencies. Officially there is no option for learning a second background, but the Skilled feat is almost worth as much as a background, and if your DM is using the Background Proficiency, I don't see why they shouldn't allow you to call it a second background.