I would still be shocked that a professional would not need to be involved for pulling a permit for a custom-designed structure.
It's going to vary significantly by jurisdiction, but for a significant majority of the geographic US....nope professional involvement not required even on a custom build. Now there are exceptions....lots of CA you can't place the portajohn and perimeter fence without a 19 page site plan and environmental impact study...but that's not the norm....at least outside of CA its not the norm
I'm in engineering/technical sales support for one of the major window companies, my territory used to be the west half of the US but I've moved over to a distributor now so my territory is now smaller. CA, parts of Colorado, Seattle to Portland of the PNW were generally the most onerus....outside of that....pretty open. Some locations may require a soil survey due to unique soil conditions, or require a P.E. stamp on a wall taller than 12' but in those conditions all they are reviewing are those specific elements. I review more then 10 fully custom house plans a week that are 5k sq ft or above and I'll probably have less then 3 a month with PE or Architect stamps anywhere to be seen. I can only think of 2 houses here in the last year that were completely designed/overseen by a licensed architect...one of which is 15k sq ft and under construction, the other is around 7k sq ft and starts in the spring.
I also hate the practice of not involving these professionals more often on these projects. The software the designers are using to make these doesn't do structural well and we end up with a lot of really stupid stuff. The designers are being paid to help meet the clients functional and aesthetic goals....builder is responsible for the structure and how it's built which includes code compliance. Unfortunately a lot of the time they just sub it out to the framer without review, the inspector(if they show up) spot checks that as built matches or exceeds the plans....and so on and so on it goes
A lot of these builds are in the range of $500 to $800 a sq ft depending on location and details....but assuming 700/square ft on 5k sq ft....hire a damn P.E. at least to review structure, headers, load bearing beams etc. You're spending a couple grand for insurance on a $3 to $4 million dollar investment. But regardless of the logic, with the current affordability challenges in housing, attempts to require additional professional involvement (which will cost money) even in the 7 figure builds will be met with significant push back.
My team gets one complaint a month from new home builders whose customers are having problems operating their 12'-16' sliding door. So we go out, pop a laser or stringline on to show the bow at the top then ask the builders rep if they want us to pull the trim so they can see the bow in the header themselves. They decline, then go back to the office and get to share the message that their 2 ply lvl header(in a 2x6 wall) was significantly undersized and needs fixed. So they get to cut drywall back, get the header straightened, add another lvl ply (assuming that alone is sufficient), then fix all the drywall, paint, trim etc. it's not fun for anyone and we try to highlight potential problems early on...but can't force them to do anything
I've made it way up the chain without completing the PE; obtaining that only has downsides for me.