Unlikely, too risky. Even the slightest glitch in the network could be disastrous. The car has to be at least smart enough to pull over and stop safely.
When talking about computation being a power drain, one could imagine a scenario where certain computations that can deal with the latency of starlink internet could be offloaded by default to the cloud. When a network failure is detected or results take too long to come in, the on-board computer could still take over the same computations, resulting in more power drain for a short moment.
There are 4,000+ planned starlink satellites that are planning to be deployed in total. This, in addition to backup with cell towers in case starlink goes down really makes this scenario risk-free.
