All those people skilled enough to get hired in the US (for massive increase in wages) don’t try to get similar positions in India, thus, nobody has to compete to pay for them.

I don't think so. You can argue emmigration takes away supply in the labor side. Why would prices go down? Quite the contrary. I don't think it necessarily raises salaries in India though, because that market seems to have a hard cap somewhere around 36k/year but it sure does opens up positions for newcomers.

Because it surpresses wages in the US, so Indian employers do not need to offer as much compensation to keep local workers who are considering emigrating.