It's quite true that means testing has an efficiency cost. One of the best ways to improve the efficiency of social security would be to convert it into a UBI.

But that's a very different question than whether it would lower the budget, and we're talking about programs that are paying out a lot more than $20. If doing means testing means you can stop paying $1000+/month to someone who is already a millionaire, that's still a savings even if it adds $20 in overhead. Meanwhile we're already paying the cost of doing the means testing, because we do it in reverse, and removing that would increase efficiency and lower spending.

Moreover, other taxes require keeping track of that stuff regardless. You already have to track the value of your assets for the purposes of capital gains tax and property tax. Doing that calculation to begin with isn't free, but the incremental cost of copying that line from the other tax forms onto the Medicare form would cost far less than it does to pay benefits to people who don't need the money. And it also has an efficiency benefit whenever it isn't a cash payment, since insurance is a moral hazard -- if the government is paying for something then you take it even if you value it at a third of what it costs, whereas if you're paying your own money you don't buy things that cost more than they're worth, so having less insurance coverage for people who could afford to pay out of pocket increases efficiency.