Scale that up by a factor of millions across the country, and you've closed the income gap. That's the power of inflation.
We don't need to get terribly cerebral to note the basic mechanism of proportion, whereby $10 added to the daily food cost of a person who makes $50/day is vastly more impactful than the same $10 added to the daily food cost of a person who makes $500/day.
Also these numbers are super weird. It's probably more like $10 total daily food cost going up to like $10.30, and going from making $30/day (part time minimum wage) to $50 (near full-time minimum wage).
I'm not seeing a great deal of causation in your theory. The same amount of food is being produced, thus the same amount of work is done - so extra employees are not required. If anything, in your closed system theory, the food producer is going to laugh and pocket extra profits.
https://www.nzherald.co.nz/business/the-conversation-inflati...
Show me which countries have wage growth for lower socioeconomic demographics currently matching or outpacing CPI? It is not happening.