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So, who's playing politics with reauthorization of the Violence Against Women Act? Sen. Schumer and the Democrats, or Sen. Grassley and the Republicans?

Well,
probably both. Yes, Democracts would love to bolster the narrative that
Republicans don't care about women, even though Grassley et al. object
to new provisions added to the VAWA, not the existing law. And yes, by
threatening the whole bill based on objections to small parts of it,
some Senate Republicans (not all of them) reveal that while they may in
general favor services for domestic violence victims, it's not exactly a
top priority to them.

Of course both Senators Chuck are playing politics. That's their game, especially in leap years.

So, the Blunt amendment got killed in the Senate. And good riddance: you wouldn't know it from the L.A. Times's writeup, but the measure was a good bit broader than a reversal of the Obama administration's contraception mandate (which itself would have been nothing to celebrate). From the amendment text (pdf):

A health plan shall not be considered to have failed to provide the
essential health benefits package...on the basis that it declines to
provide coverage of specific items or services because...providing
coverage (or, in the case of a sponsor of a group health plan, paying
for coverage) of such specific items or services is contrary to the
religious beliefs or moral convictions of the sponsor, issuer, or other
entity offering the plan.

In other words, essentially a line-item veto of whatever the boss is morally opposed to, based on church teaching or otherwise.