Democrats are not center-right, and comparisons to European political parties are not accurate. The US has a “first past the post” electoral system. US political parties are more like coalitions of smaller parties.
The Democratics encompass all the left wing parties, except the Greens (who are essentially Russian agents at this point). The Republicans are the conservative coalition (immigration nuts, rich dudes, Nazis, people who have experienced a trauma and are taking their anger and fear out on the world, etc.).
If you want to move the Democrats left, you need to get elected and raise money like AOC does. Sorry, shouting on the Fediverse doesn’t count.