I don't think you'd lose every last Republican, because this still does give many Republicans what they want... an end to "Obamacare" while actually finding a way to keep most of its benefits. One thing we've learned is that, even in the reddest states, they may hate the fact that "Obama" is in the (colloquial) name of the thing, but they actually like what they're getting out of it. So a solution like this is also a way to save face. "See? We ended Obamacare! And we're offering something better!" This should play well for many Republicans at home.
Yes, there is a group of Republicans who have always wanted Repeal with no Replace, because they think the federal gov't should get out of health care altogether, and they will certainly vote against something like this, but that is a minority. The whole idea of "repeal and replace" came about when Republicans discovered that, in fact, their constituents actually want a good healthcare program. Paul Ryan knows this. I think he, too, would love to find a way to pass a "repeal" and even he must be coming to the realization that, with the Republican party as fractured as it is, he's not going to be able to do it without getting Dem votes. He'd like to save face, too.
This will not put private insurance companies out of business. I pointed out some ways they could compete, and in fact we already have plenty of examples where for-profit companies can still compete with non-profit entities. Education (at all levels) is a good example, as is the fact that UPS/fedex compete with USPS.
They could also set it up (much like the ACA) so that businesses (over a certain number of people) would continue to be served by private companies, and only individuals (and employees of the smallest businesses, who as today, would not need to provide coverage for their employees) could buy into Medicare. The point is, the pie is huge, and there is room to keep private insurers happy as well, while still moving in a more Dem-friendly direction.