When it comes to basic staples/commodities where the vast majority of volume is sold as store brands, wages have little to do with prices. It's about supply, cost of getting supply to market, and lack of substitutes.
On the branded side, I think lack of competition has been the biggest driver of cost increases. PepsiCo's gross profit and operating income the last 12 months is higher than it was in 2019. If you want soft drinks in a can or bottle, it's basically Pepsi, Coke, or Keurig/Dr. Pepper brands. All of the M&A activity the last 5-7 years had decreased competition to the point where most industries are oligopolies if not duopolies. If you want bran flakes with raisins for breakfast, you usually have at most three choices - Kellogg's, the store brand, and maybe Post. If one raises prices, the other two follow, and then so do you unless there are adequate substitutes.
I used to make eggs almost every Saturday & Sunday. Haven't bought eggs in at least 6-8 weeks; going with my Mon-Fri breakfast of oatmeal, a banana, and a protein shake. But whey protein costs have shot up too; the brand I prefer used to be $45-50 for a 5lb tub, now that's $62+ even on Amazon subscribe & save. So my substitutes have increased in price as well.
Few people actively look to spend more money. Wages may make a difference in mid-to-high end brand usage, but not basic staples/low end of the cost spectrum products.