You can see some comparison data here, or find other sources (I don't have any bookmarked to dig up, I last did the research on this a few months ago...) https://www.restaurantbusinessonline.com/financing/why-big-m...
They'll jack up the prices for any reason, don't get me wrong, but it's not because they have to pay people more. It's because they see it as an excuse to pay themselves more.
The people controlling the price of the sandwich are also the ones who want to pay their workers less.
We shouldn't cower and run at the thought of the bosses using strategies to fight back by comically jacking up their prices and then cynically blaming it on the government.
It's to be expected.
I don't know in what world a third is "only a small part". The same article mentions According to PayScale, the average worker in the fast-food industry makes $8.50 per hour. Increase that to $44 and you end up with a much more expensive burger.
[1] https://marketrealist.com/2019/11/how-mcdonalds-wages-major-...
From the 2018 annual report: https://corporate.mcdonalds.com/content/dam/gwscorp/nfl/inve...