Interesting idea, but much depends on how one judges "doesn't benefit everyone". For instance, housing subsidies can reasonably be described as benefiting everyone. How? Obviously they benefit the people getting the subsidies directly because those whose wages are too low to otherwise afford decent housing can then afford some approximation of decent housing. Obviously they benefit property owners in areas where there is a demand for housing but the costs of building/maintaining housing outpace the amount that can be paid by low wage earners ... thus helping make it economically viable for rental property investors to build properties to meet the need for housing. Less obviously, they also benefits all of the customers of companies that employ low wage earners, as it helps those companies keep their wages low and thus avoid raising prices for their goods and services.
Do you ever shop at a store that employs people for low wages? If so, you've almost certainly benefitted indirectly from housing assistance.
Further, housing assistance, food stamps, etc. reduce desperation, which helps avoid things like riots and property damage. That's good for anyone who own property. Do you own property? If so, then you've at least indirectly benefitted from housing assistance, food stamps, etc.
Do you hope to retire? If so, then you can reasonably expect to benefit from social security and Medicare. (Although you might not benefit at today's levels, assuming that the politically insane continue to thwart efforts to re-balance the income limit and continue to thwart things like universal healthcare that -- if done even vaguely close to half as well as most of the rest of the world -- should reduce our percentage of GDP/capita spent on healthcare ... which would certainly benefit all of us.)
Speaking of healthcare, healthy workers are a huge boon to employers ... they don't have to worry about their skilled employees being out so much. That's a giant monetary benefit to employers and by extension some benefit to all fellow employees as a result of the employer being more stable because of the healthy workforce.
So, what exactly is this public institution that supposedly doesn't benefit everyone?
We live in a very interconnected society. Short of hermits, I don't think you'll find anyone who isn't benefitted by most every program of any significant size in some way ... even if only indirectly.