"At any rate, I think your assessment is a more depressing version of the American body politic because it means a small minority of fickle swing voters are the cross-section that both major parties seek to gain and thus have all the power. However, this does not disprove my argument that the root of gridlock in Congress is the need for two similar parties to appeal to a shared constituency."
Perhaps, but our electorate grows more polarized each year. Most of the major issues have been sorted out, so most people naturally gravitate toward one party or the other because they have fairly cohesive messages. Nonetheless, the share of people who are neither Democrat nor Republican also grows each year, primarily due to dissatisfaction with both parties, not as a matter of ideological differences, but as a matter of disgust.
"Because Republicans and Democrats are big-tent parties, there is no complete uniformity across their respective membership."
There is moreso than you might think. Sure, there are a handful of people in any group who will hold beliefs outside the norm. That number is shrinking each year though. There hasn't been a mass mobilization of the United States which led to a great shared common experience since World War II. Vietnam came close, but the culture wars of the 60s, including preferential treatment with respect to selective service/"the draft", shattered any semblance of cohesion, never mind that there was never a declaration of war, so we technically never lost the "police action." We've had generations now for the Dems and Reps to grow further apart.
As for "big tent", that used to be true, but now it's just a catch phrase both parties use to try and make them seem appealing to a wider audience, though their positions are set in stone.
"How meaningfully different are the resultant votes in Congress? For example, I don't think I would be able to tell the partisanship of a given Congressman if he gave me his stance on gun control."
I kind of think you would, though gun control might be the most difficult to distinguish because the second amendment has broad political support. Every so often the leaders of congress throw up test votes for their caucuses. These votes, especially in times of gridlock, are not intended to actually pass. However, they do two things: 1) give friendly politicians a chance to vote their conscience so they can campaign on that issue with their constituents come election time, and 2) force opponents to vote against the legislation, again to campaign on the position during the next election.
One thing that I will say though is that our system is strictly first-past-the-post by district. We don't apportion any seats via proportional representation which are then handed out to party hardliners. I'd say that type of system creates more intraparty vitriol and hatred because there is jockeying for position for a prized seats in parliament. In the United States, we re-elected 96.4% of our Congressional representatives. That's because our districts are by and large set up to give a huge advantage to the incumbents due to widespread gerrymandering, especially so after the 2010 census and ensuing redistricting.
I do feel some sympathy for the "fickle swing voters" though, especially if they live in Florida, Ohio, North Carolina or Colorado. That very small sliver is really bombarded with political advertisements and mailers during presidential elections.