The Overton window is a measurement of acceptable political discourse. You shift it by vocally expressing unpopular public opinions in the public sphere, not by voting. Voting is supposed to be an expression of popular public opinion, not the other way around.

Sometimes the most effective way to change things is by threatening not to vote and convincing other people to do likewise. The most famous use of this tactic was by the civil rights leader Martin Luther King Jr.

In the 1960s black people were much more actively discriminated against on a systemic level, practically prevented from voting in many of the states in the southern United States. The president at the time was the Democrat Lyndon B. Johnson, and was facing the much more racist Republican challenger Barry Goldwater. While the black vote was suppressed in the south, there was a significant voting block in the north of black people and their allies whose main issue was civil rights. Civil rights leaders, including Martin Luther King, met several times with LBJ, who coaxed them to tone down the direct action protests and criticism until after the election, as he claimed to we willing to negotiate with them once the threat to his power was diminished. Instead, civil rights protests increased. The leaders, probably correctly, determined that once the election was over, they would have less leverage. Even though losing the election meant having an enemy in the white house, having a 'friend' who continued to delay essential concessions did not further their cause. People were actively being murdered by the 'Jim Crow' apartheid regime, and delays and half-measures were not sufficient.

Thanks to the pressure of millions of people engaged in direct action and open criticism of the president, the Civil Rights Act was passed before the 1964 election. LBJ won by a landslide due to the popularity of the legislation, but suffered the severe political consequences Democrats were trying to avoid through their strategy of placation and delay. The 1964 election was the last where Democrats got the majority of the white vote, and electing politicians in the southern states became much more difficult for their party. Palestinian Americans and their allies now face a similar situation. Democrats will continue to ignore the genocide in Gaza unless there are real political consequences to their actions. While Donald Trump would be a significantly worse candidate, the logic of a two-party system requires that they be willing to risk a worse political situation if they are to hold any political power at all.

Civil rights leaders like Martin Luther King are regarded as heroes today, but at the time, they were reviled by both Democrats and Republicans as a force of chaos that acted out of ignorance of the political system. If LBJ had lost the election to Goldwater, perhaps their legacy would be considered differently. But it would not change the fact that the cause they were fighting for was just, and they were able to wield political power in a system that was designed to marginalize them.