It only works if hiring H1Bs is cheaper, or otherwise much more appealing, so that paying the premium on an auction makes sense.

It used to be the case, say, 15-20 years ago. It not need to be the case today. Since we're talking about big tech, let the minimal bid for a company be the median salary across that company's relevant line of work (engineering, sales, whatever). This would make hiring an H1B candidate a merit-based decision, not a cost-cutting measure. This would make hiring a US-born engineer and, say, an India-born engineer approximately equally expensive, so the company would hire the better engineer, not the cheaper.

If the price arbitrage were gone, I bet there'd remain enough H1B slots to invite better researchers, better flute players, better sea captains, etc.