I own a contracting firm with 12 in-house crews working 50 hours a week, 52 weeks a year, as well as a plumbing company. I deal with more permits and inspections in a month than most people will deal with in a lifetime. The idea that permits, licenses, and inspections make work better is simply laughable. What they mainly are is an industry protection scheme promoted originally by union plumbers and electricians and now continued by all the licensed trades to keep outsides from cutting into their profits.
First, most building inspectors are just about useless. They are typically guys who failed in the trades and decided they would go become an inspector. Can't tell you how many inspectors I know who "used to have" a plumbing business, electrical business, or whatever. I've never met an inspector - and I know dozens of them - that I'd trust to frame a new build, plumb a bathroom remodel, or wire a room addition. If they were any good, they'd make way more doing the work than inspecting the work.
Second, there's no way they actually understand the entirety of the IRC, so they all develop their little picky things that they will call out on every visit. Most of these little picky things have nothing to do with overall quality. So contractors learn, "Oh, this is in Bill's zone. Be sure to add extra expanding foam in all the corners, but be sure to miss one so he can have something to point out to you. Then ask him about his fishing boat." Doesn't add a thing to the overall quality of your job. It's just part of playing along with the legal game.
Sure, they can recognize really terrible work and may save someone from that. But with REI groups, Angie's List, HomeAdvisor and all the other referral sources out there, the only reason you'd be hiring a terrible contractor is if you picked the absolutely lowest bidder without regard to references....don't do that.
As for licensed contractors, I have seen truly abysmal work performed by licensed contractors - not just tradesmen or journeyman, but guys with master's licenses in plumbing or electrical. They're good at passing a test. Maybe not so good at installing plumbing so it doesn't leak. Texas, where I live, doesn't have licensing for general contractors. We've had a huge construction boom here for years, even when the rest of the country was in the dumps. If we've had a rash of houses falling down due to lack of GC licensing, I've missed it. Meanwhile, some states feel they need to license painters. Your tax dollars at work, protecting homeowners from a bad paint job. (Not really, the licensed guys still often do crappy work.)
People say permits don't cost that much. That's simply not true. The cost of permits is not just what the city charges you. It's that you're then required to use a licensed electrician to do simple crap like move a vanity light, or install a can light. And that licensed electrician either reports to a master electrician, or uses an RME (Responsible Master Electrician) to pull the electrical permit, and that guy wants to get paid. So the light that your handyman could relocate for $50 now becomes 3-4x that.
Want to install a new tub and plumb it? Any decent handyman can do that work in a day, probably at $50/hour or so. But pull a permit and now you have to have a licensed plumber, who's got a master plumber upline to pull plumbing permits, do the work at $150/hour. Except you still need your handyman, because God knows a plumber isn't going to actually install and caulk the surround on the tub.
Permits kinda make sense for really large jobs, where contractors might be tempted to cut corners. They definitely don't make sense on the average bathroom or kitchen remodel, or other typical cosmetic work.
Now all that said, would I trust a $40K job to a handyman? Probably not. But different people use that term in different ways. In our area, it means "a guy who does mediocre work in a bunch of different areas." But I hear in other areas, people actually use it with pride. So maybe your guy is one of those.
All that said, if you've had multiple reputable contractors - emphasis on reputable, make sure they're that - tell you they'd skip a permit on your job in that particular area, with that set of circumstances, I'd be a lot more likely to listen to them than a bunch of internet strangers (me included).
