This attack lets you use all the functions of the key fob, and not just the action captured.
Someone looking to break into your car will probably use a brick, not a flipper zero.
https://i.blackhat.com/USA-22/Thursday/US-22-Csikor-RollBack...
Suggests that it can be used to start a car. Whether it was a fob start or push start isnt specified.
... It's literally in the title on the first slide.
It is only an attack on keyless entry. Keyless start systems all use a different short-range radio system.
An attack on keyless entry might let someone trigger a remote starter if supported, but this doesn't enable anyone to drive the vehicle, because you will need to authenticate with the keyless start system after entering the vehicle before you can put it in drive (and to keep the engine running for longer than the predetermined timeout).