in general, reasons not to means-test:
1. What you said: admin costs spike, sometimes more than savings.
2. Paperwork and bureaucracy involved means that some "deserving"/eligible people don't get it because they mess up or forget the paperwork, or because the test is insane.
Some extreme examples of this from the UK's means-tested disability benefits over the last decade, the same happens with US Medicaid and disability payments, and also school meal vouchers in the US. Also happens in India in massive amounts for our govt food programs.
3. A small means-test (excluding billionaires, say) can be expanded quite easily (excluding all above the official poverty line).
4. Universal programs (the NHS in the UK, Social Security in the US) create a wide base of support that keeps them politically untouchable for at least decades. It's easier to gut means-tested programs that only benefit a small proportion of the population, especially if you can demonise that segment and that segment's voters are captive to one party.
5. The same thing (clawing back money from high earners) can be achieved by universal programs and progressive taxation.