Obviously the number, length, and complexity of flights would have a big impact as well. For 1 or 2 round-trip domestic flights per week, it should be doable.
Source: Had a friend who worked on several private flight crews that operated like this.
Edit: They mostly worked with clients in the $100MM/yr range, who tended to treat their staff like garbage and consequently had high turnover. Über-rich lurkers, YMMV.
The billionaires I've met, who by and large own and don't rent, have what I'd call three distinct use cases for private air travel: business, social, and family. Business travel can range from more or less daily to at least 3x per week. Family travel is less frequent and generally limited to whole-family trips, but said trips can and do occur a lot more frequently when you have those means. And then there are what I'd call social trips -- a bucket into which I'd lump anything from charity events, to fundraisers in DC, to golf tournaments, to spur-of-the-moment trips to the house in Aspen or the Hamptons.
Said billionaires, if asked, would consider most of the above to be business trips, because their business lives never stop, and some measure of business is usually baked into most trips -- even if it's a quick stop at one city to conduct business before heading on to the vacation spot.
There aren't very many of these jobs going around at any given time, and they involve exhaustive security and background clearances. You are also privy to a ridiculous amount of inside info flying these guys around everywhere, and you need to be trustworthy, i.e., come with references from other billionaires and such. It's a very hard world to break into, and if flying commercial paid better than this did, no one would bother trying.
That's worth a lot to a billionaire.
A pilot has your life in their hands. Smaller jets can be very 'shifty' when landing which is scary. Also you want a pilot that can land anywhere.