You can't possibly tease out all the different causes of the effect you're discussing here. Are you sure the worker shortages weren't
due to the inflation? Much of that was greedflation, where companies raised prices due to the crisis and then kept them high, but wages didn't go up, so some people - especially those with long commutes - were actually working at a loss.
Not to mention that the amount given - a total of $3200 to childless people over an entire year - amounts to around 3 months of an average 1 bedroom apartment's rent in America in 2020, much less in cities, and not including groceries or utilities. As a young, childless person who got the full amount that year, it gave me exactly what I needed to be able to move out, but certainly wouldn't have sustained me - even in a "flop house" - long enough to really avoid looking for work.