Markets aren't perfect. Very few people claim that they are. Markets don't solve all problems. What markets can allow you to do is to allocate resources in the way people want them allocated. In some ways it's democracy in action. Except you only get as many votes as you have pound notes, so it's not all that fair.
Take education as an example. Everyone knows there are good schools, OK schools and a few truly crap schools. Talk to the parents in an area, and they all know which is which, and which they really don't want. At the moment you broadly get into the good schools by living near them. So house prices are higher in the catchment area. Obviously it's more complicated than that.
But in a lot of cases, you get what you're given. There's been an attempt by government to have inspections, and try to force change on some of the worst schools. Which has had mixed success. It's a big old unweildy system, and there's a lot of competing interests pulling in slightly different directions.
Another approach might be to give all parents a voucher, and let them spend it where they will. With safeguards. It could be chaos at first, and this might make it unworkable. However, there's a chance that it might lead to a more responsive system. And fewer kids might get a chance at a decent education that otherwise wouldn't have been open to them.
One advantage of this, is that we could re-open the village schools. If parents wanted it, they'd pool their vouchers. It probably wouldn't be enough to pay for it alone, but with use of say a free church hall, some volunteer help and top-up from parents or the parish council, it could be possible to have a village primary again. Which would be a decent example of local democracy in action, something this country could do with.
To be honest I'm not sure about education vouchers, and whether they're workable. But there's a perfectly valid argument for them. Monolithic state services tend to get into a habit of saying "you'll get what you're given."
What I think would benefit from change is the NHS. Take Belgium as an example. You get a better service over there, and it costs them less money. As a country they're far to the left of us, and yet their health service is semi-private. The state and universities run the teaching hospitals, there are private ones, state ones, charity ones, union ones and company ones. Your GP will point you in the direction of where to go. Ten years ago they were doing heart bypass surgery for 1/3rd the cost of the NHS, with better survival and post-op infection rates. And virutally no waiting lists. My friend ran a company bringing UK private and government patients over for treatment. They took 1/3rd and spent it on hotels, travel, translation, hand-holding, form-filling and profit for them, the UK patients or taxpayer saved 1/3rd. Admittedly I wouldn't go for the complex mutuelle and top-up insurance system that pays for it, I'd go with taxation and vouchers. But no-one complained that GPs were never nationalised in this country, and have always been private contractors to the NHS. I think a mixed system would be much better, and could be achieved piecemeal as well.