Actually, 51% of the votes might get you anywhere between ~10% to ~90% of control, based on how that 51% is distributed. In PR, if x% vote for Party X, Party X holds x% of seats simpliciter. In first past the post, it's distributed by constituency. If you win 49% in every district and 99% in the single most populous district, you get only one seat even though you got strictly more than half the vote (assuming a strict two party system).

One interesting difference between the US system and the parliamentary system is how coalitions work. In one sense, US parties are themselves coalitions already. Democrats are a coalition between labor, feminists, environmentalists, minorities, leftists, civil libertarians, and various single-issue voters; Republicans are a coalition between militarists, free-marketers, right-wing Christians of all denominations, and various single-issue voters. If you're a gun-owning trade unionist, who are you going to vote for? In practice, places with a lot of gun owning trade unionists end up with pro-gun Democrats and pro-union Republicans, and then they have to dynamically form coalitions just like a parliamentary system. Likewise, you don't get huge factions of the Labour party voting against a Labour PM in Parliament unless they want to start something; whereas you do get that in the US system. There are even bipartisan coalitions, between conservative Democrats and Republicans during the Reagan years, or even single-issue coalitions across both parties (though these are diminishing--"pro-life" Democrats and "pro-choice" Republicans have become increasingly ostracized and purged from their respective parties, for instance). So in effect, the US system allows dynamic coalitions to form between individual members of Congress, while a PR-based parliamentary system (theoretically) allows dynamic coalitions to form between multitudes of small parties. (In practice, there are still pretty tight upper bounds on how many different parties you can have, for reasons best described as "brand recognition".) What's the effect? In a parliament, it's harder to form a coalition, but once formed the coalition effectively runs the table until someone decides to start over again. In the US system, every individual piece of legislation can pass or fail, and it's an errand to shove it though.

Gridlock happens for two reasons: one, the US system is designed to gridlock, two, because opposing sides are learning to fully utilize this design rather than cooperate against it. The US system is designed to gridlock by having a bicameral legislature (most parliaments are effectively unicameral) with an independent executive and veto. The rules of each house require bills to pass through committees (and, in some cases, cloture) before coming up for a vote. Then you need an executive signature, or else you have to go through the legislative system again with higher thresholds for passing. Additionally, each house, as well as the executive, is elected separately. Unless each house is controlled by the same party as the President, you have to compromise to get anything done (or, alternately, one side can choose to completely obstruct the other). Even if each house is controlled by the same party as the President, you have to get everyone within the party to cooperate.

"With no political parties, every single vote would be about political dealmaking and organizing politicians. We would never get anything done."? Every single vote is already about political dealmaking and organizing politicians. That's how the US system works.