The roads are administered by the local governments. A lot of the time it is not the local residents who are using these shortcuts.
Either way, the town I grew up in had a number of "No through traffic" signs. Generally in areas where people would cut through a neighborhood for a right hand turn instead of going through a light-controlled intersection.
There is a moving violation in many jurisdictions called "Avoiding a traffic control device" that allows officers to ticket people who cut through parking lots and alleys to avoid traffic lights, etc. Laws that restrict the use of smaller roads are consistent with this philosophy.
Regardless of who pays for the road, residential streets are not designed to handle the larger traffic volumes of major streets. The road surfaces are not as durable as major roads, and traffic control devices that are safe for a residential area are not safe for higher traffic loads.
People are not entitled to use things just because their taxes paid for some proportion of that thing. You can't show up at an armory and ask to borrow a gun to shoot some squirrels. Without rules and incentives regulating the use of public goods, public goods are abused, leading to the tragedy of the commons.
This might seem like a design issue, but you can design residential streets to be poor shortcuts by making them curve all over the place and installing traffic calming measures, except these measures also typically make neighborhoods less bike-and-pedestrian friendly when compared to normal gridded streets. One-way streets and traffic restrictions work just as well though. You also avoid the issue of entitled drivers honking every time they go over a speed bump or are obstructed by a pedestrian they think is taking too long, or a USPS truck that blocks one lane every day at the same time.