Of course, but the government has commited prices (>£100/MWh) for wind. You have to pay them at least that to compensate, otherwise they won't stop producing. Which is going to get extremely expensive quickly.

You can stop windmills. However that changes the investment nature of wind farms. There are a lot of ways to structure this, and no one answer so it is hard to type up all the different issues. I'll pick one simple case, but don't assume this applies to all.

An investor (group) gets a contract to supply wind to a utility for $x. This price is pretty low since wind is variable, but it is enough to predict over a year how much they will make and so for the investors it works out.

For the utility they need more green energy in the mix, and so they are willing to sign the contract even though sometimes they will buy electricity and they pay someone to take it on the spot market. They also know how much wind they expect to get, and so those times wind is negative are made up by times that wind is blowing when they would otherwise have to buy much more expensive peak plant power. So for the utility over the course of the year the wind power is profitable even though at some moments it is unprofitable.

The problem is in the above separation the windmill owner is paid for all power produced, and so if they shut down there is no money for them. (or they have to agree with the utility on a fair payment)

This sounds like the issue railways had in UK 100 years ago. Solution was to nationalize the infrastructure. If the nation would own the wind-farms and nuclear plants, both can be managed to the benefit pf the nation.

If we privatize essential infrastructure it means -> yea the owner will want to have biggest profit possible and will not be flexible vs rest of the grid.

National railroads don't have a good history in the US. Though here it was the cities taking over.