Your perspective is that both parties share blame, but yet you also admit you're not particularly well-versed in the recent (20 years) history of immigration legislation.
You want it to be one way, but really it's the other way.
In
2005 and
2006 the House and Senate, respectively, passed immigration bills. The House version, passed with 92% of Republicans in support and 82% of Democrats opposing, was a set of draconian anti-immigrant measures. The Senate version, authored by a bipartisan set of Senators, passed the Senate 62-36, with 39 Democrats and 23 Republicans in support and 33 Republicans and 3 Democrats in opposition. It was a more even-handed set of measures. Since the two bills could not be reconciled, nothing ended up being passed.
In 2007, at the urging of President Bush, the Democratic majority Senate tried a compromise bill of the 2005 & 2006 efforts which eventually failed cloture votes.
In 2013, the Senate passed a
bipartisan bill on immigration 68 to 32, with all Democrats & Independents, and 14 Republicans voting for the bill. The House, however, never took up the bill due to threats from conservative Republican legislators. The next year, sitting GOP Majority Leader Eric Cantor lost his primary (the first sitting House Majority Leader to lose his primary since 1899) in part due to his open-ness (not even support) towards immigration reform.