If it were me, I would make companies "bid" on the visas, where their bid is the salary (including all benefits, minus ALL expenses) they will pay the employee.
Employers who really need employees for high-skilled work would thus get visas. Employers who want to underpay their foreign labor will not.
The auction or bid for H-1B would lead to only Big Tech being able to hire them.
Other uses of the H-1B (e.g. French teacher https://h1bdata.info/index.php?em=&job=french+teacher&city=&... ) wouldn't be able to compete. There are also places in rural America where the only way to get doctors and nurses ( https://h1bdata.info/index.php?em=&job=nurse+practitioner&ci... ) is to hire them from abroad (the other option is to close the facility as insurance reimbursements aren't high enough to pay more).
Auctions are great because it's harder to game than a lottery. If a company truly believes there is a shortage of rocket surgeons and they need to access to the global market of these specialists, they can bid high. No hand wringing about not being able to find American rocket surgeons (willing to work for low income country pay scales). No stuffing a lottery with tons of entries - bids incentivize applicants to not waste slots. No worrying that a Federal department made the wrong decision on total supply, price, or who the recipients are - auctions self-adjust and update.
Auctions are bad because need is not entirely measured in money. Highest bidders win is not egalitarian. Sometimes we do want to subsidize rocket surgeons visas for disadvantaged parts of the country.
My concern with such approaches is around the "all skilled labor is lumped into the H-1B visa", be it needing fluency in French and a teaching certificate or a PhD in data science with a focus on economics - its the same pool of visas.
We keep saying H-1B. There used to also be a H-1A visa and an H-1C visa. https://en.wikipedia.org/wiki/H-1A_visa
What if, instead of trying to solve the H-1B visa problem from companies that have gone through the roof for visa applications ( https://en.wikipedia.org/wiki/H-1B_visa#Top_H-1B_employers_b... ), instead split off all the jobs that are classified by the BLS as 15-0000 ( https://www.bls.gov/oes/2023/may/oes150000.htm ) and move them to a new section ... Lets call it H-1D.
Within the H-1D which is for tech jobs (just like H-1A and H-1C were for healthcare while they were active), it can have a different pool size and requirements and for that one could require a certain base pay and have an auction on top of that.
https://www.uscis.gov/archive/h-1c-registered-nurse-working-...
This could allow for regulating tech job immigration differently while minimizing the impact to French teachers or medical professionals in rural America that have no way to compete with Big Tech salaries.
The reason I think this is more likely doable is that it works the same way that the H-1A and H-1C visas did before to address one section of the skilled labor visa applications.