They have done that with all the ones that could be done already. I'm assuming any domestic worker at this point has to be physically here for some reason. Security is a big one that comes to mind.
That would be a very poor assumption. Almost all work can be offshored. If the need is high enough, the electorate will even elect people to remove government regulations requiring onshoring. Any job can be offshored. People will travel to other countries to get surgery if the cost difference is high enough. Residential home building and maintenance is a rare exception, but even commercial plumbers can be offshored if the entire business is moved offshore.
At any time our economy maintains a certain equilibrium until a large change occurs. Sometimes that is a recession (net immigration was necessary during the housing recession) and sometimes it is regulatory changes like the one mentioned in this article. If you make H-1B more expensive, there will be winners and losers. There will be some people hired in roles that would have been filled by H-1Bs, but the research is overwhelming that overall employment of citizens will drop. A combination of less local consumers, increased offshoring, and reduced competitiveness of US firms will cause that lower overall employment.