This does not mean that Rs won't get their own little zinger. I predict the following:
a. The House bill will include a provision that prohibits the Administration from requiring that any religious organization, or any for-profit or non-profit enterprise affiliated with any religious institution, provide contraception as part of any health insurance plan. It will also prevent the Administration from doing so indirectly. For example, it will allow such employers to request that any insurance plan explicitly not offer contraception either for free or as part of a covered plan. Religious employers (and affiliated institutions) will still be allowed to offer such coverage, but the government will not be able to require them to offer it and will permit them to prohibit it on any offered plan.
b. The provision may or may not get stripped in the Senate. But even if it does, the House will put it back in during conference and it will pass as part of the bill.
c. The Administration will sign the bill, because they are not going to crater a central part of their economic plan over this issue. The Administration will lament how Republicans have used the political process to deny women needed coverage supported by most Americans, and will console itself and its allies that this at least doesn't make things worse. It merely preserves the status quo.
Needless to say, Progressives will generally be pissed at this. And they will have a point. But it also highlights the current weakness of the reproductive rights movement. That, unfortunately, is a more global problem that will require significant effort and strategizing to correct. In the short term, however, it is extremely difficult to block when there are so many "must pass"* bills.
*For those unfamiliar, "must pass" is a term of art in legislative politics. It refers to a bill that everyone generally agrees "must pass" to avert some extremely unpleasant outcome. Of course, it is always possible for a President to veto a bill, so "must pass" is not literal. As always, I refer the curious reader to Heinlien's novella "Magic, Inc." (usually sold with the novella "Waldo" as "Waldo and Magic, Inc.") It contains the best fictional account I have yet seen of a legislative session and how lobbying is actually done.