Let me put it this way. You seem to be saying that because more people were working there was more money in the economy. Which is a fallacy. You might as well say prices went up because sellers felt like it. It would at least make mire sense.
It wasn’t just more people working, but more people working per household. When you take household income and increase it by 50-100%, yet there are none of the purchases of homes and appliances to go along with that, you have more money floating around in the economy. There was more money while the expenses were about the same for a little while. The increase was in disposable income. That’s what made inflation possible.