What you would get would be a lot of back-room deals and strange bedfellows. Like now, only more so.
That's not to say that any bill that gets enough bipartisan consensus to pass a 2/3 vote must be good; plenty of bills that have passed in the past with that much consensus have been bad. But I think it might change the dynamics in at least something like the right direction.
Make everything 2/3 vote and you increase the need for the dealings that override the current 2/3 vote tools.