I believe this to be true overall. McDonald's or bigger places wouldn't hire less people, if they could get away with lower staff levels they already would have lowered them.
For small business it's one percent or less change per employee.
For small business it's one percent or less change per employee.
To zeroth order, that seems reasonable. But the added expenses might shift the cost-benefit analysis enough to add motivation to explore automation or other changes to reduce the workforce. The necessary R&D or capital costs might have not been attractive with lower wages, but could become attractive with higher wages.
Or that the self-service machines are going to arrive regardless of the wage, with the only difference being McDonalds keeping workers on $10/h wage 3-5 years longer than they would keep the $15/h workers?
What happens when the entire self-serving machinery costs as much as it does to pay one or two employees for a year? Does it make sense for McDonalds to keep human employees at any wage?
On the other hand if people like the machines more they might raise prices.
Automat restaurants have existed since the 1890s and were first touted as a cost cutting measure.
Ironically enough, they were most popular around the 1930s and disappeared roughly when the minimum wage was highest (1970s) and are now making something of a comeback, when the minimum wage is again, historically very low.
And something as simple as checking straws and napkins or making sure the floor by the soda machine is clean can't be automated cheaply. Automation in general is too fragile to handle the demands of that kind of work.
"The system" was so good that it revolutionized the fast food market to the point that almost every attempt to buck the trend is now considered some sort of synonym to 'craft' or 'artisanal'.
(Note: I'm taking the movie as canon, which is basically stupid, but I have a grandfather who would gladly tell you the story of his first McDonald's visit, and how revolutionary it was relative to everywhere else he'd been, but for his passage a few years back.)
And we already have a model of what it will look like, laundromats. That's what people are aiming for, the patron doing a lot of the work with maybe one person to oversee and provide special services like dry cleaning. I don't think that would be an improvement in food service.