Solving the filibuster
There are three ways a president can beat the filibuster: win 60 seats in the Senate, negotiate with the minority party, or work on little things that both parties can agree upon. The first is impossible, the last is rare now, and Biden has chosen negotiation with Republicans, though Mitch McConnell's obstructionism makes this difficult. This leaves us only with backdoor, rule-bending measures like budget reconciliation or cloture.
So, how do things actually get done? Through executive actions and White House power; through state and local governments and courts; through social movements like the Capitol riot; or, they simply don't get done unless they're power grabs.
The Senate-president institution is incapable of getting work done. But there is a solution, it's surprisingly simple, and brought up often.