it could be tough to defend the name in litigation, and you might not be able to do anything with others who used the same name before you did.
That's the part I contend though. I understand perfectly that Bill's Pizza is not a strong brand, but not for the litigious reasons you put forth. If you have your company name registered, that's it. You need to expressly defend your trademark in order to keep it, but as long as you have a trademark registered in that particular trade, nobody can touch you. If another Bill's Pizza comes around, you have your attorney send a cease and desist letter, and go to court if necessary. What it sounds like you're talking about in the legal world is trademarking singularly common names. That can't be done anyway. In other words, I can't trademark "pizza" but I can trademark "Bill's Pizza." I can't trademark "deli" but I can trademark "Stone Deli." Apple cannot be trademarked, but Apple Computer and Apple Records can."get employees to come up with non-descriptive terms for products and services"
I still think you're using the term "descriptive" while intending to instead say "singularly common." Bill, for example, might be common, and pizza might be common, but there might not exist a "Bill's Pizza" anywhere. The trademark office will decide for you what's defendable and what isn't. This shouldn't be a legal issue.
Bill's Pizza is a poor choice of branding because of the backyard scale of operations it connotes.