Regarding cars, some families offer a monthly transportation stipend in place of a car. So it's possible to work around this. It's completely dependent on your location and the AP's preference.
All of our APs eat dinners with us. One of our most independent APs, whom we hardly saw on weekends, actually preferred to eat dinner with us most nights. She'd get take outs from time to time, but saw dinner time as us including her to be part of the family.
Having an AP will mean some loss of privacy. It's to be expected. We've had APs ranging from very independent (never hangs out with outside of working hours) to always joining us for activities, including after dinner family board games and movie nights. Both personality types work with us. If you prefer to have an independent AP who does her own thing, you can certainly screen for that.
Many APs will ask during the interview whether you have pets. I've had several APs tell me that though they love dogs and have family dogs themselves, they prefer not having to take care of a host family's dog. It's not their job and they fear the dreaded mission creep. I don't blame them. If you want your AP to take the dog out midday on a regular basis, offer extra money for that task.
The biggest benefit of having an AP is the consistency of coverage. You don't have to scramble for coverage when the kids are sick and can't go to daycare/school. Or when daycares shut down every month due to covid exposure or staff shortage.
The corollary is you have to find an AP who's generally compatible with your caution level and who won't go partying in crowded bars every weekend.
The biggest downside is it takes a lot of mental energy to manage a young woman living in your house. It's a balance act to have high expectations of her (as a child care provider and a roommate) and ensuring that she's happy too. The last thing you want is an unhappy/moody person living under the same roof.