I'm not sure if some American companies have this. I don't believe they do. From hiring I know for sure, if there is someone who doesn't need immigration handholding is about as qualified as someone on an H1B, the former is preferred. Most of the times there is explicit guidance - "we aren't hiring H1B's from this date until further notice". So I know from being on both sides of the coin (as an H1B and then not needing one but being on the hiring side) there isn't a preference to H1Bs. In fact I'd assume the reverse is true.

Now, H1B's will put in longer hours and extra work without complaint and won't take things like EEO action, legal action, etc. against their employer. But it just comes with the territory that as a visitor in the US you do not want legal trouble and would like to preserve your legal status as seamlessly as possible.

To your first point - the H1B exists because they can't find technical (or other) talent in their work zone. So here's the thing - if one company can't find it, neither can others unless that one company is doing something super specialized. They could just provide the ability to move in zones where that expertise is needed and the minimum expected salary the employer must pay. There are solutions if congress gets off their rear-end and tries to find them.