Whether you leave blue tooth on all the time or turn it off when not needed (as I do) the car still needs to wake up it if is asleep.
Perhaps when I get a "real phone" I will leave blue tooth on all the time.
Most of the time the issue is not the car, or the car sleeping. It's not seeing your phone because people have power save on which kills some background things in the phone that prevent it from working. Or running some venders buggy "customized" version of Android.
Can you connect to your car when it's sleeping? Well most people can. There are many levels of sleep in the car.
My car is "sleeping fine". If it's only using 2 miles a day while parked, that as deep as I expect it will sleep. While still handling phone as key and answering an LTE phone connection remotely etc.
My iPhone is constantly "seeing" the car in my house (I'm 80 feet from it now and it shows an active "Phone as Key" connection). But it's Bluetooth LE that uses milliwatts of power. It doesn't mean the door will unlock. It just sees it. Lot's of stuff in the car are always on. LTE, Bluetooth Radios, Key Card Radios, Some Battery Monitoring etc.
There are dozens of ways the protocols "Wake up" and go to a higher state. Like I said you are overriding the whole purpose of "hands free" phone as key. The phone is usually the problem. But you've just "rebooted" Bluetooth by turning on Bluetooth and woke the phone from a deeper sleep. When most folks have issues it's because that deeper sleep hands free protocol fails, but you bypassed most of it.
On an iPhone if you turned on Bluetooth, it would wake every app registered for background with Bluetooth. You want that to all work without doing that sort of thing. And that's where it gets tricky. That it all works without wasting power. Some don't trust it can still shut their Bluetooth off.
I've written both iPhone and Android apps that have to connect to BLE devices just like the car. iPhone was a breeze. Android what a nightmare. Because ever vender hacks up Android differently, especially around power management and sleeping. Samsung got it right 2 years after Pixel. If you want "official" Android, get a Pixel.