The only solution here is if the ride is absurdly high, to the point where you break even with drivers in the nearest high-population city, which people also don't want since they'd be paying multiple hundreds of dollars for a ride. At that point, people will just get a car.
The issue with uber is that it's compensating for a lack of public transportation that takes you exactly where you want to end up at (or public transportation at all in most of the U.S.). Maybe the only way car-on-demand is profitable is if (A) we get self-driving cars, or (B) the government creates their own system with lower fares and runs it at a pure loss with no profitability in mind.
If Uber can't operate while paying minimum wage then maybe they shouldn't be operating.
Every other company has to work out how to pay minimum wage. An unprofitable rural convenience store doesn't get to pay its clerks less because the sales aren't high enough, so I fail to see why it should be different for Uber and their drivers.
To be fair, an appropriate Uber setup here would also require that drivers know the destination beforehand and get compensated if the ride is cancelled and have the ability to decline rides.
The problem for both babysitters and uber drivers in low-usage areas is that there is NO economic model that allows for 40hr workweeks once population (or usage) declines sufficiently. All these gig models work very well in the city but totally break down as you go to rare use areas. A new model is required -- i'm not saying Uber is it, but Uber is closest.
Anyone who isnt in an urban environment, or who's landed at an airport at 2am knows this is a problem. We just need a way to fairly compensate people to solve it.
I.e in your example, where someone wants to work in 30 minute increments and that is the staff member's choice and the employer is happy, then fine.
However it would be unreasonable for an employer to say to a staff member "I'm only going to pay you in 30 minute increments, and after each 30 minutes you can go watch TV, but if a customer arrives when you are on break I want you to run back to work straight away to staff the till, and I'm not going to pay you for the time watching TV".
It's reasonable to ask the Babysitter to work part-time and have fairly short shifts. An example of being unreasonable would be to say that they have to take calls during the day with customers and they will only be paid the minutes talking to customers. It's worth saying the laws here change if you are an employee vs sole-trader, and most babysitters would not be classed as an employee while Uber drivers are.
For any situation in-between, the answer probably ends somewhere on the spectrum between "yes" and "no".
When i'm in rural areas, there are often no taxis and one will show up an hour after you call, maybe. Without order liquidity, it is infeasible to maintain supply. Which company (or person) would stand ready to ride just in case an order came thru once every 4 or 5 hours?
The current market response to this is simply not supporting the market. The alternative is Uber where presumably the person is doing their yard work and jumps and does a ride if one happens to pop up. I cannot imagine Uber will sponsor idle wages in rural regions where you get an order or two a day.