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).