If you can substitute foreign labor then there is no shortage. If you can't substitute foreign labor then it's irrelevant.
You can't have a shortage of workers and falling wages. You can have uncompetitive US companies, but they are very profitable so the only thing left is irrational behavior.
Or insufficient capital investment and bad management holding down productivity. You can't pay a worker more than their productivity, but if the best you can get out of them is too little, it may not be worth their while. You could then plausibly see a shortage of workers, wage suppression, and stagnant productivity statistics at the same time.
But that can not be reconciled to "labor shortage".
A more fruitful line of research focuses on the growing power of monopoly, and the collapse of new business formation in the USA. Why don't companies with huge profits reinvest their profits? Why are most of the tech giants sitting on piles of cash? Monopoly could explain that, and would also explain a limit on wages that otherwise seems to defy market sense.