Some H1B workers are paid less. Some are paid more. Using the company's median wage for that role is not a reliable measuring stick. "Prevailing wage" is a better metric of the labor market's pricing for labor, and does take into account job title as well as geographic demand for that labor.

If "A total of 60% of all H-1B jobs are assigned wage levels that are well below the local median wage" that is actually not a terrible stat. 50% are above the median and 50% below among US Citizens (because of math). 60% is worse, but you have to dig more into the stat to convince me. Also, how far "below" is dictated by the prevailing wage ranges (which will closely align with general labor-market pay ranges.

"Underpaid" immigrants under an H1B are easily recruited away (via visa transfers to a new employer who won't underpay them)--something I've done frequently. [But now people won't be happy that we're recruiting them and paying them well!]

I think you're looking for a bogeyman without considering a breadth of factors (you should also consider the legal costs of processing and maintaining an H1B and other factors transporting and housing some of these workers if you want to talk straight-dollars).

If we workers want to capture the labor market we'll need to do what our employers work every day to do: Do more and do it better, for less. Sorry--welcome to the most successful economy in the world.