If I understand the totally dysfunctional US political scene with even approximate clarity, the Senate would certainly block any attempt by the House to pass more adequate legislation regarding women's reproductive rights because the Republican Party would filibuster.

Furthermore, several Dem Senators would demand various concessions in order to vote in favor, which would further reduce the chances of any bill being passed.

The problem isn't any one political Party; the problem is that the USA has always had a totally corrupt and dysfunctional political system; today it's simply more obvious to more people.