If the power station is making a profit then why would the generating company want to shut it down?
In a lot of cases they aren't making a profit.
If you run a coal fired station then you are hit from all sides.
To start with there are ever tightening emissions restrictions which are causing expenditure on plant upgrades (flue scrubbers and the like).
And then you are forced to pay money into the shakedown pot to pay for the ROCs and FITs to renewables operators.
And you have to compete in the energy market against all the other generators - including those you are subsidising.
If you want to run flat out for long periods, you have to bid very low and make no money. If you don't bid low then you only get to run when the cheaper options aren't sufficient - and that means lots of starts/stops and power variations which is hell on big machinery (causes lots of wear and tear, and hence high maintenance costs).
And then there's the biggie.
If the politicians have finally woken up to the realities, and the graphs are showing a huge shortfall during winter, then ... you announce that you are closing your huge power station. TPTB then go "oh sh*t" or whatever the official terminology is, and decide that your plant is too important to security of supplies and will pay you "availability payments" to keep the plant available.
On the question of stuff not showing up in the figures ...
The likes of Gridwatch and BM Reports only report what is metered onto the grid and therefore visible to the energy market (that's what the figures are being reported for). There are a lot of small generation installations which are not metered at the grid level as they are only connected into the DNO network. So as well as roof-top PV, there are a lot of small hydro and wind installations that don't appear in the figures.
Given the number of them, the small unmetered wind installations add up to quite a bit. I recall seeing some figures a while ago, but I can't remember what they were.