The most important factor in the organization of Congress is very simple - political parties. Without the parties, which form themselves into conferences or caucuses, Congress would be a very different place. I am not aware of any national legislature, parliament or Congress that doesn't have party organizations. It is the fundamental glue that keeps the organization intact. That does not mean that they have to exist. While it is rare, it is not unheard of for assemblies of some form to forbid partisan organization. School boards, city councils and an occasional state legislature have (or have tried) non-partisan approaches. Even in such circumstances, party ties usually lie just under the surface and exist informally. A former Speaker of the Minnesota House of Representatives told me that when the legislature in Minnesota was non-partisan, it really didn't make much of a difference. People still coalesced into party-type groups anyway.
The idea of getting rid of partisan labels is a noble idea but is akin to a principal of a high school decreeing that all student cliques are abolished. They will exist in other forms no matter what the law or the principal says. There is an overwhelming negative view about political parties that is as old as the American Republic. The Constitutional framers were keenly worried about factions and parties. They were unable to stop them, as they quickly joined their own factions to debate the Constitution itself. If you don't like political parties, I would suggest you find something else to dislike, because you are not going to get rid of them. Even Communist regimes held the single party as the ideal, but it still was a party.
The party organization in Congress is the crucial organizing and disciplinary element of the institution. You could function without them, like Minnesota did until the 1970s, but they would exist in a different, more hidden form. The party that has the majority of seats organizes the chamber. If there is a tie, which has happened in the Senate recently, then both parties must negotiate a settlement. Negotiated settlements in the U.S. Congress are normally very rare, but in other parliaments they are common-place. The current Canadian parliament currently has a minority government where the governing party does not have a majority of seats. This happens when there are more than two parties in the chamber. Congress generally has had just two parties (with an occasional eccentric member claiming to be an independent or some other designation).
The parties in Congress also wield influence over members. The powerful party whips will corral wayward members to vote with the party. With what can you be whipped? Usually it is just the honor of the party, but it could be money for your next campaign, it could be favors in the form of legislation, it could be a threat to support a challenger in the party primary election. U.S. Congressional discipline is fairly mild compared to other national legislatures, but the glue that holds parties together in the U.S. Congress is fairly strong, so that severe disciplinary measures usually are not needed when discipline is required.
There is a big difference, however, between a party in Congress and a national party organization. This is true everywhere. A congressional party organization is normally an effective little club. The national party organization often is not. The relationship between the congressional party and the national party is not hardwired. The Republican National Committee does not control the Republican Conference in the House or Senate and neither does the Democratic National Committee. In fact the parties in both chambers have their own partisan organizations to organize campaigns with the state party organizations - Senatorial Campaign Committees (Republican and Democrat) and Congressional Campaign Committees (Republican and Democrat). The National Committees, though normally friendly and sympathetic, are separate groups. The National party claims a genuine interest in the party's fortunes in Congress but is really more attuned to the Presidency and the states.
Any political scientist will tell you that political parties provide a critical role in the development of public policy in a democracy. But when you have political parties you inherently have partisanship. I have frequently heard, and have even moaned, about the extreme partisanship that now plagues Washington, D.C. But these complaints are just like complaints about the weather. Unless you can honestly cite a genuine change in the climate, complaints about the weather and hyper-partisanship need only wait awhile. Extreme partisanship has cycled through Washington, D.C, for over 200 years. The early Congresses were obsessed with Democratic/Republican vs. Federalists or Democrat vs. Whig or Republican vs. Democrat. Early Presidential campaigns would make cable TV viewers blush today. Things have always been intensely partisan, and at times the intensity became red hot. An entire section of the country seceded from the Union because a Republican won the 1860 election. (No, I am not exaggerating too much).
Partisanship ebbs and flows along with one factor - the margin of the majority. In the early part of the 20th century, Republicans frequently ruled with large majorities in Congress (and usually the White House), and partisanship was less intense then. In the 60s, the Democrats ruled with large majorities, and partisanship was also less a factor. When the parties start alternating or battling precinct by precinct in the states. That is when extreme partisanship really began to heat up. People who yearn for the good old days of bi-partisanship are really just yearning for a time of a large single party majority.
The current Congress is very closely divided by party. The Democrats control the Senate by only one vote. The past two Presidential elections have been famously close. Expect extreme partisanship to continue into the foreseeable future. Is this a problem? I would argue it makes no difference. Bad legislation can come from monolithic single party controlled legislatures, but I am at a loss to think of really good legislation that has ever come out of very closely divided Congresses. It is just too difficult to get anything of substance passed in an extremely partisan season. The current debate about immigration reform is a good example. It is too big an issue for the current Congress, as it is currently configured, to handle. Do not despair, however, since eventually one party will gain the upper hand and partisanship will decrease. Then the big party will get corrupt and the other party will replace them, followed by more partisanship. You can see the cycle. Congress and parties are all very human.
There is an element, however, that the normal weather pattern of partisanship did not have to count on in the past. A sort of climate change in partisanship may be at hand. Previously, partisanship cycles in the past were fed primarily by newspapers (if you think newspapers are biased today, just look at newspapers of the 19th century). Broadcast television and radio were rigidly controlled by the fairness doctrine in the past, and were never really a factor in fueling partisanship. Cable television, uncontrolled by rules and offering a much broader selection besides the old 3 ½ networks of yesteryear (yes, PBS counts only as ½), now has more viewers than broadcast television. Radio, now unfettered by the fairness doctrine, is alive with opinions. The big culprit, however, that may mean a very long spell of extreme partisanship is the Internet. A myriad of blogs now opine on everything, fueling and even creating issues no one even thought could be partisan. Who can complain, however? Citizens now have more avenues to express their fundamental right of free speech and some argue that means democracy has now been restored to the people. Should we block the Internet and censor speech in order to try to calm partisanship down? Good luck. We might as well relax and enjoy the party!