You're mistaking talking about it with the press being unable to ignore Trump like they'd done with the previous politicians who'd made a fuss about it, modulo the symbol laden example of Disney pulling this stunt, which, you'll note, they never made a "campaign" about. To take one of the most ludicrous and iconic example of this, the NYT had much more important things to do, like get women membership in the Augusta National Golf Club.
It's also the case that H-1Bs only seriously affect our domain of "IT", and unless you're in that, it's not in your face like Latin Americans e.g. transforming your neighborhood. Something I witnessed in the dozen years I spent in central Arlington, VA (although these, at least, made good neighbors. Heh, in the D.C. Metro area Arlington was by a good measure the most "diverse" of the municipalities/counties).
Per Wikipedia, it started in 1990, thanks to G. H. W. Bush and the Democratic Congress (https://en.wikipedia.org/wiki/H-1B_visa). Which would have followed the NSF's et. al. 1980's campaign to reduce the cost of scientific labor by claiming there was a grave skills shortage.
As for "suspending it", Sessions at least took the "Intel yes, Infosys no" approach with his 2015 bill, but I agree the whole program is a problem. But I'll take his and Trump's position, which I think will take legislation, as a good first step.