The problem is that through traffic often drives above the speed limit.
B) If speed is the issue, that's fine - address the speed. Increased traffic is not the issue in this case.
I don't know if it's cheaper per vehicle mile to drive on a large road or a small road, but I couldn't see it being very significant, but maybe it is?
Increased traffic on a road not meant to take it indicates some unmet demand, but it doesn't necessarily follow that the correct response is to accept the traffic on that road. It may be better to open things up elsewhere.
Also note that it's possible for adding routes to reduce efficiency, and thus for removing routes to increase efficiency: https://en.wikipedia.org/wiki/Braess's_paradox
Finally, there's the problem of externalities. Road traffic has costs to the people nearby, in terms of noise, pollution, and danger. The drivers don't experience those costs, and rarely factor them into the decision of which route to take. Putting a lot of traffic onto a residential road not meant to take it can easily be a net negative.
It may be better to open things up elsewhere, as long as the reason that it is elsewhere is not because elsewhere had a weaker residents association.
I've seen the paradox you mentioned a few times, but I think it is like a lot of paradoxes, interesting but rare and not very relevant most of the time.
It may be a net negative for the people on that street, but overall, I'm not so sure.
Additionally, there is the safety aspect. It is more than just pedestrians at risk, there are other automobiles and structures to be concerned about. Tertiary streets, for example, probably won't have things like guardrails.
Areas zones as residential should probably not have much through traffic. It is expensive and dangerous. The roads constructed to deal with that traffic are also built to withstand that higher load. Routing the traffic elsewhere will not significantly reduce the maintaining of said roads.
Err... I modeled traffic. Hopefully that explains it. There's some good write-ups as complaints about Waze, if you want a keyword to find more info.
What I don't know is if the cost of a small road per tonne-mile service life is any different from the cost of a large road per tonne-mile service life. If the cost difference is negligible then I don't see the problem.
Maybe it is that different parts of the government fund the small roads than the large roads, and the parts of the government that funds the small roads doesn't have the budget to do much road work?
Do heavy vehicles generally use these small streets, it seems like it wouldn't be worth it in a heavy vehicle with most small streets.
The cost differences are huge. A primary road will have been constructed to tolerate the weight and frequency. It's much more expensive to maintain such a road.
In some cases, yes. Maintaining and improving roads will be paid out of different budgets. Federal, State, (sometimes) County, Local Municipality. Tertiary roads are almost certainly out of the local budget.
If it's not expressly prohibited, there's bound to be heavier traffic on these tertiary streets. The heavier vehicles do far more damage.
(There's also safety to be concerned with.)
It's okay to route traffic through these areas when it's an emergency. Occasional use is not going to be a significant impactor. However, with regular use, it's going to result in rapid deterioration.
If you really want to dig into this, here's a link to a study:
https://www.lrrb.org/pdf/201432.pdf
There are many, many more. If you're curious about the weight issue, it's known as ESAL and this page is pretty easy to digest:
Maybe a better solution would be to put in barriers.