That’s not really how it works. It would only make them look stupid if they did that, and increase the risk of being voted out in the next election. Because of the filibuster, legislation generally needs 60 votes in the Senate before it goes to the president. Democrats have 51 seats, with two frequent defectors. Imagine if Biden’s record going into 2024 is that he vetoed every single piece of legislation.
I use https://searx.be/ with both Google and DDG results enabled. Otherwise if you want something like Google results you would have to use Start page, although they pay Google for access to their search so you would be indirectly supporting Google.