I'd actually worry more about how efficiently the money is being used after it's been disbursed, not about how much government administrative costs eat up.

E.g. that chart says that ~96% of Medicare spending went to beneficiaries. That sounds good, but all it means is that 96% of the dollars were paid to hospitals/doctors/medical device manufacturers/etc. It says nothing about how efficiently those people are providing their services, i.e. how efficiently those 96% of Medicare dollars are being spent. A lot of those dollars get quickly chewed up by high prices (which I'd argue are themselves caused by the market distortions that these programs foster).