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".
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.