As to the speed limits and driver's license laws those are for public roads. You can drive a car without a license as fast as you want if you own it or have permission and are on private land. They do not interfere with you property rights(what you can do on or with your property). There are also laws as to what can be driven on a public road but you can drive anything you want on private land.
animal bans to interfere with property rights, what I can have on my own land and what I can own.
Property rights have little to do with the driving laws you mentioned. Those laws are written to specifically apply only to public roads and land. While you have some rights regarding your property, they are not universal. Depending on the wording of the law, it can apply to public areas as well as private property. If you think you can do anything you like on or with your property, you are sadly mistaken.
For instance, if you live in a city, you may be required to cut your grass, or refrain from watering during a drought. Obviously, you can't rob or murder someone just because they are on your property. Likewise, you can't deal drugs or make a meth lab on your property either. Zoning ordinances can even tell you whether you can build on your land or run a business on it.
Laws regarding animals are no different. The City, State and Federal Government can all tell you what you can keep and how you must keep it. I certainly don't want a meth lab next door to me, neither do I want my neighbors 12 year old son to keep Gaboon Vipers (or lions, or chimps, etc). He couldn't take care of them, the animals would suffer and they would also be a risk to me and my family.
I'm sure price is part of it but you could also get a captive bred bird that was not hand raised for less as well. Seems odd someone who is not smart enough to know the difference between hand raised and wild in quality of pet is smart enough to find and get involved in a smuggling ring of animals to get the cheapest price.
To many (otherwise) very smart people, the price is more important than the origin of the animal. They think they are smart enough to overcome any problems that might arise. If you want breeders, you might prefer smuggled animals for their bloodlines and the fact that they aren't handfed.
Ultimately, depending on what species you are looking for, it's not all that difficult to find smuggled animals.
All would be totally a mute point any way if you just made the law that captive bred animals need a ID of some sort and the person keep proof of sale.
It is also a point to consider that banning them in captivity would not stop smuggling and would likely increase the efforts as there would no longer be a captive bred alternative.
There is no 100% foolproof way to ID captive bred animals. Paper ID's and receipts are easily forged. Even it there was an attempt to legislate it, you'd run in to people such as yourself that argue "property rights!". It would be extremely difficult to get everyone to pay to get their animals marked, chipped, tatooed, etc. ...and the government wouldn't pay for it. What you ask for is just not reasonable in this society, at this time.
However, contrary to what you may think, banning them in captivity would GREATLY reduce smuggling. In many cases, wildlife officers may know the animal was smuggled, but proving it is impossible. As long as there is a legal market for some animals, smugglers will flourish. If you ban the animal, then you no longer have to prove it's smuggled, it's mere existence here is proof enough. Further, by banning an animal, you reduce the market for it. If it gets small enough, there is not enough demand to make large scale smuggling profitable.
Look at Ivory as an example. If it's banned, then it's easy to confiscate all you find. That reduces the market for it and slows the killing of elephants. However, if you start opening loopholes that let some into the market, then how do you tell which ivory came from legal sources and which came from poached animals?
Years ago, while there was a ban in place, elephant poaching was greatly reduced. Now that some countries (China and Japan for example) have partially lifted the ban, poaching has shown a huge increase. It's now at some of the highest levels we've ever seen. That doesn't bode well for the elephant.