‘Mildly wild’ is a frankly weird term, and not one that really means anything. If we’re talking about non-traditional and non-domestic pets then there are lots of different aspects of their care that must be considered to keep them well, and I am a little concerned with how easily available many of these species are.
The way I see it is that a truly domesticated animal is adapted to living with humans in their environment. A non-domestic animal is not. The human environment needs to be adapted to keep the non-domestic animal.
There are some species that can be kept as pets, but the set up and time required for their ideal care is almost more like running your own personal zoo.
(As a side note, there are some species which should never be kept as pets in a domestic setting, and large, social carnivores spring instantly to mind. USA, your exotic pet laws are weird. You can get toucans? Owls without a license? Cobras for goodness sake. Who’s interest is that in, really?)
I mean, it’s relatively easy to keep a dog or cat and meet all it’s nutritional, physical and social needs. They’re not set and forget like a houseplant, but in the grand scheme of things dogs and cats are fairly easy to introduce to a household.
Compare this to a reptile where it really needs its own specialized environment within the human environment to be kept adequately.
Compare this to a parrot, which not only needs its own environment, or modification of the existing environment, it needs significant time and effort dedicated by its human towards enrichment and activities.
Keeping the more exotic pets often falls somewhere between ‘petkeeping’ and ‘zookeeping’. All pets are a commitment, but some are more so than others. I would not be able to keep a parrot, I simply lack the time and a suitable environment to do go.
If someone is going to keep one of these species as a pet, captive bred individuals is greatly preferable to wild caught. Captive bred individuals are at least somewhat acclimatized to the human environment, and have not affected wild stocks.
Unfortunately many of these non-domesticated species are cheap and easy to acquire, and owners who are pursuing a novelty factor rather than dedicating suitable time and effort to that species often end up with them.
The Discworld series with always be my number one, but I will always have a soft spot for the Animorphs.
Keeping pretty much any non-domesticated animal is akin to zookeeping. Some animals (smaller lizards, dart frogs, small fish) are happy in small enough enclosures to make keeping them practical with research and responsible care. Others (green iguanas, toucans, small lemurs) would certainly be possible to keep, in their own room-sized enclosure, but would require more dedication, time, money, and specific equipment than most people are able or willing to provide.