I changed location from South Bay to San Francisco within the same company and they had to file a new LCA and an amended H1-B. It basically is the whole process again, minus the lottery. (Since I already had an H1-B, I'd already been counted against the yearly quota.)
If you create a bidding war and localize it, the H1B shops are going to exploit this even more.
Except that minimum salary thresholds for H1B's are already set (for each region). We're talking about just increasing these thresholds (e.g. 55K --> 85K and so on...).
Now we have to talk about cost of living adjusted thresholds.
But in general you're correct that employees can change work sites while the H1-B petition is pending (because without premium processing that takes months), but only after the new LCA is approved. And the LCA is what determines the prevailing wage for the new site. So you can't move someone from Bumfuck, Nebraska to San Francisco and continue paying Bumfuck wages.
If you think the processing time is too long - I totally agree. 15 days should be regular processing time, not "premium".
I think many of the "nightmare" issues are already present and already being managed just as well (or badly.) The marginal addition in complexity shouldn't be that high.
Some small nit-picks as an example: H1-B's are not locked to a specific location, so it would be easy enough to bring in employees for the Kansas City office, then immediately relocate them to the Bay Area at the same salary. Changing this will be really hard since free travel across the country is a fundamental US right.