That doesn't solve the problem, which is OK, in that I don't expect people here to understand how being on call works.
Basically, it's like a leash centered on the hospital: You can be anywhere, as long as you can get to the hospital in a certain amount of time when you get called in. This roughly correlates with distance, but not entirely: When my dad was on call in a town with a railroad track running through it, he had to stay on the right side of the tracks because a long coal train would blow his response time. (It was a small town.) The point is, though, they aren't fixed at home when they're on call, and being fixed at home would destroy their lives. My dad had to be on call fully 50% of the time when he lived in aforesaid small town, because he was 50% of that town's total anesthesia providers.
Therefore, unless you propose a chauffeur, they will be driving their own vehicles while they're on call.
On a deeper level, changing how being on call works would require changing how small of a town people are allowed to live in. This seems like a common problem: People come up with something that they might be able to make work in New York City, and think they've Solved The Problem, blissfully unaware of St. Louis, let alone Du Quoin.