It still wouldn't fix downward pressure on American wages, and it also increases the negotiating leverage of employers vs. their H1B employees.
If employers bid for the permits themselves, there'd be a large fixed-cost component to employing an H1B, but then no incentive (indeed, an extra disincentive, since the company had to shell out to get them to the U.S.) to pay them well afterwards. And since H1Bs are sponsored by their employer, they have limited job mobility and can't just go work for a competitor, particularly since the competitor would then have to shell out even more money to win the auction for the visa. That'll put downward pressure on immigrant salaries, which in turn will put downward pressure on American salaries for workers who do the same job, as employers could substitute a one-time fixed cost to "lock in" an H1B rather than paying prevailing market wages.
It would reduce downward pressure on wages, because the permit cost would be bid up until there was an equilibrium (based on the present cost of all potential wage savings on the future). This would serve two purposes; first, there would be more transparency, and the number of visas available could be adjusted to find the 'right balance', and second, employers would be most willing to pay a high premium for talents unavailable in the domestic market.
To be clear, I think all these restrictions are terrible, and am an open borders advocate, but find the design of these programs very interesting (and near impossible).
If employers bid for the permits themselves, there'd be a large fixed-cost component to employing an H1B, but then no incentive (indeed, an extra disincentive, since the company had to shell out to get them to the U.S.) to pay them well afterwards. And since H1Bs are sponsored by their employer, they have limited job mobility and can't just go work for a competitor, particularly since the competitor would then have to shell out even more money to win the auction for the visa. That'll put downward pressure on immigrant salaries, which in turn will put downward pressure on American salaries for workers who do the same job, as employers could substitute a one-time fixed cost to "lock in" an H1B rather than paying prevailing market wages.