The Left ultimately wants to abolish government power and create a stateless, classless society. Leftists differ among themselves with regard to the value of government power in achieving this, with the two opposite poles of the discussion being anarchists and Marxist-Leninists. But in general, leftists realize that they have nowhere near the organizational power to even think about those goals, and just want to improve the material conditions of working-class people, who make up the majority of society.
The Democratic Party is far from being controlled by leftists: I can count the number of actual leftists in the US House on one hand, and in the Senate on one finger. The Democratic Party is a center-right party, dedicated above all to the stability of the existing ruling class. They sometimes virtue-signal cultural values that are shared with the left (feminism, anti-racism), but largely as a means of providing an outlet for those values that is not threatening to capitalism, and which can be co-opted to protect it.