We're seeing with Mamdani, who was elected with a massive mandate, that he can get things done despite a hostile state government that clearly didn't want him to get elected. Obama had a mandate, yet he didn't use it. Looking back, we should have seen this coming when he was mentored by people like Joe Liberman, and we immediately saw the results when he was tested in one of his first goals (healthcare). He dropped the public option immediately and implemented a Heritage Foundation plan(Yeah the Project2025 heritage foundation).
You can do a lot when you have a mandate and you are determined to get things done. How to determine if a candidate is like this? After many cycles of election in my experience there is only one reliable barometer. The only determining factor seems to be money in politics. If your candidate has corporate contributions, you can pretty much guarantee that nothing is going to get done. On the other side, we've seen non‑corporate‑backed candidates actually try, at the very least.
Now, talking about the actual specifics of how you get past a gridlocked Congress and a hostile Supreme Court: with the advantage of being at the top of the party, you have the ability to whip your party into falling in line. We saw this with FDR. He wasn't pulling any punches. If someone got in his way, including the Supreme Court, he made their lives miserable. One of the reasons bernie people feel fond of him.
A lot of these politicians are only there for the corporate donations and in hopes of a job afterwards. So a very low‑hanging fruit is to go after the corporations that are donating to the politician who is holding things up. Make the public aware that this SOB is the one holding up plans that are very popular. Do what Trump did: go into that person's district and get his voters to realize that this is the person that's holding everything up.
For example, paid family‑leave polls show ~71% overall support, including Republicans, Independents, and Democrats. Yet when Biden campaigned on it and then immediately dropped it at the first sign, it showed that he didn't even try.
A lot can be done if the candidate actually tries.