So the obvious way to improve things is just to adopt more House-style procedural rules. Strengthen the leadership vis-a-vis the individual members, make it more majoritarian, etc. The House is hardly perfect, so this would hardly solve all problems, but it does solve many problems by—in essence—cutting down on the number of parties needed to negotiate something. Stronger leadership both makes it easier to do partisan bills without silly buy-offs, and also makes it easier to forge bipartisan deals by getting the key players into a room. The highly individualistic nature of the Senate is anachronistic—a legacy of a different period in American life. We have a partisan politics these days, and what we need are political institutions that fit that paradigm.What's worth pointing out here is that while the House has not always had strong party leadership and been run by the majority party, it always has had stronger leaders and weaker individual Members than the Senate. During the bulk of the twentieth century, the strong leaders in the House were committee chairs, not party leaders -- but they were still strong leaders. In fact, the reforms of 1959-1975 didn't just shift power up to party leaders; it also shifted influence down to subcommittee chairs (and away from committee chairs, and of course eventually strongly away from committee ranking member)..
The bottom line is that because of relatively homogeneous districts and the large size of the House, individual Members are willing to trade in their very small influence over most issues for a somewhat larger influence on a narrow set of issues. That yields hierarchical government in the House. Senators, with a smaller chamber and with very diverse interests, are not willing to make that trade. Nothing about the current partisan situation has really changed that incentive structure, and so the Senate is still going to resist becoming a hierarchical institution.
That's not to say that Senate rules are working well today; I think there's good evidence that some reform would be a good idea. But that reform needs to be designed with the fundamental structure of the Senate in mind, or else Senators, majority and minority, won't accept it. The Senate doesn't want to become just like the House, and no outsiders are going to convince a majority of Senators to make that happen.