Any side road advantageous to a cyclist (good pavement quality, shorter distance) will have cars, unless it's brand new and nobody knows about it.
You have to find the good side roads which are not usable to drivers, due to, say, obstacles that a cyclist can get around easily. If it's not obvious from the entrance to that road, you will need local knowledge.
There is always a risk that if you go down some random side road, you may be backing out to where you started.
Riding on roads outside of city limits isn't such a big deal that you'd bother, in the first place. It depends on the exact conditions. What is the visibility like? (Curve with rocky wall on one side, precipice on the other? Or field?) How wide is the paved shoulder? What are the speeds like?
Everywhere else, the pyrennes, the alps and company are a spiderweb of tiny towns with tiny quiet roads between them, in a ratio of about 5:1 against the larger main roads. I've cycled them, traffic isn't a problem.
As for paving, on a tour I'll take a hard pack, unpaved surface with no traffic over a main road all day every day.
And knowing where the roads go? That's route planning! Have a little look at where you plan to go before you set out each morning. Strava's heat maps are really useful, though understand that road racers do prefer the highway, and they cycle a lot!
I do this every year, and traffic just doesn't come in to it except at the large terminal cities where we get the international trains or ferries from...
The A-C road may not be any different from the A-B one in terms of A-B being "main" and A-C being a "secondary" road, and the path will likely be longer.
A road actually going to a neighboring town is a strawman example of a side road.
He is talking about cycling for travel, exploration, sport and leisure - not about commuting. I have just done the same last week, a 5 day tour and can confirm, I planned everything around secondary, tertiary and country / farm roads. It was mostly empty, and a pleasure. (despite being longer than the "main" road)
The Great Salt Lake in Utah is the size of Lichtenstein. The Grand Canyon of the Colorado is longer than the Italian boot is wide. England is the size of Alabama.
It can be harder to find a safe route somewhere in the country, I'll give you that.
The US is not like that. There are only a few routes in the west because the west is more than a thousand miles of mountains and deserts nearly all above 1500m and much of it above 2000m.
It is hundreds of miles between north south routes in Arizona because of the Grand Canyon. And the eastern route goes from vast remoteness to vast remoteness, Navajo nation to the Great Basin (an internal drainage the size of France).
I am not a general fan of American exceptionalism. But for geography, it holds. The average height of the Colorado Plateau is about 2000m…almost as high as the highest point in Australia. And the plateau is larger than Germany.
I am glad it works on your machine.
This is vaguely similar to Australia, which has gigantic deserts with terrible roads through them as most of the landmass, but geographically bikeable population centres where everyone actually lives.
Agrigultural roads that you'd never bother (or even want) to take your car down. Residential streets that only the residents would drive on.
I'd guess that on the average 5 day cycling tour in the UK and Europe, I'd see less than 1000 cars total, and often go for many hours without seeing any. Its glorious.