They absolutely will. The problem with just setting a high price is that they'd be leaving money on the table whenever anyone who could afford even higher prices comes through. That's what they want to eliminate.
Even if you aren't earning more money the algorithm will constantly be testing you by jacking up prices to see what you'll endure under which circumstances. Screaming kids in the car with you? You'll probably put up with an extra 38.33 cents for those happy meals. Looking tired? A recent death in your family? Do your resent social media posts indicate that under more stress? Enjoy higher and higher prices to stress-eat your comfort foods peasant.
I don't see any of the backlash Wendy's got for this kind of bullshit either.
Look, I'm all excited over the incoming cyberpunk dystopia but no they won't be able to do this. They still have posted menu prices and folks will be pissed if prices ever go up from what they're expecting to pay.
Digital signs and apps are what they're pushing on us now. There are already restaurants that don't have any kind of printed/posted menu at all and you're expected to use your phone to scan a QR code. There's no telling if the prices shown to you on your device are the same as what the person behind you in line will see.
There are also already restaurants that have different prices when you use their app which helps condition you to accept that prices will differ from person to person or even day to day.
You're right that people don't care for discriminatory pricing though. People's objection to the practice is the only thing that's been keeping it from being more widely used already (https://link.springer.com/article/10.1057/s41272-019-00224-3)
They stand to make money hand over fist from it though so we're going to see company after company testing it, pushing the boundaries of what we'll accept and normalizing the practice. Without regulations making discriminatory pricing illegal in certain settings, it's basically inevitable that we'll see it more often.
They would probably be able to do it if the listed prices were fixed, but they provided a personalized discount that would vary depending on who you are.
This is already how the app works and indeed pretty much every form of price discrimination runs this way. It's just that that the "who you are" part often isn't as obvious because you'll just encounter a coupon code and not realize that by virtue of you seeing it you're part of a specific demographic they're targeting.
But the reverse of raising prices beyond menu price depending on who you are will get pitchforks.
Well of course, except the menu prices are the "max" and everyone gets varying discounts depending on their profile. Just like at your local auto parts store, basically nobody pays "list price", yet it's there for the odd walk-in cash sale who doesn't know how these things work.
(they will just have high prices, and lower them until less wealthy can buy)