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What problem do they solve?

Not sure if trolling, but in addition to the above--it's just a better level of service for cycling. The bike shops will be glad to have fewer people popping in asking to use a pump, it's always important to tell people to pump up their tyres (and look after their chain--very common reasons why people enjoy cycling less), and it sends a visible signal that cycling is welcome and expected.

It was also found that it helps people appreciate the value of a track pump--every household should have one, but many people are unaware that they exist and have never used one.

I would much rather the money was spent on any number of other things that would be of use to cyclists. More bike racks, for example.

The budget the local authority has for on-street cycle parking far outstrips the cost of these pumps.