No, unfortunately, it wouldn't. You can't vote independent in the abstract. You must vote for human beings. Organized policial parties vary somewhat from state to state, but in most there is no organized Independent Party. In fact, if there were an organized political party that called itself "Independent," anyone who joined it would, by definition, no longer be truly independent politically speaking.
This means that even if you register as an independent, when you go to the polls you'll need to vote for someone from one of the organized political parties.
In general, your choices are limited to:
Republican
Democrat
Libertarian
Green
Constitution
Natural Law
Conservative (in New York State only, I believe)
Communist
There may be a minority party I've missed, but you get the point. Something around 96% to 98% of Americans who go to the polls vote for Democrats and Republicans.
The only practical way to have even a remote chance to change this dynamic would be if considerably more than 50% of the American electorate could be convinced to change their registration from any organized party to independent. That way, the two dominant parties could no longer make assumptions based on the number of registered voters they have district to district and state to state. Polling firms would be forced to come up with a different model, one that is not primarily based on the number of Democrats vs. Republicans.
Of course you are right, but wouldn't it be great if We the People really could set these arrogant self-serving asses on their heels!