Not really... there's some problems with your hypothesis.
First off, it depends on what you mean by variability.... if you mean the difference between high and low temperature each day, which is what your graph shows, that is indeed greater in the summer most places (exceptions are out there, check out a monsoonal place like India).
Whereas variation can also mean the range of how hot or cold each day can attain, which is often greater in the winter (see how tall the line or red areas are in the graph below for London):
So each day typically has bigger movement in the summer, while the winter tends to have wider swings year-to-year and often week-to-week (depending on when and where fronts and storms go)
But focusing on the actual diurnal difference, which does indeed tend to be greater in the summer... there are problems with some of your thoughts on moisture:
Clouds are evidence that there's quite reasonable moisture in the air... they form when the air moving to that level is "full" of moisture. They don't rob the air of moisture, they're evidence it has "too much".
And indeed there is more actual water vapor in the air during the summer in the UK, as measured by the dew point temperature:
(Source for values: TimeAndDate.com)
- Water vapor does absorb some sunlight (more than most tropospheric gases)... but is not the primary way the Earth is heated:
(Source: NASA education site)
The ground absorbs most solar energy, and then heats the air in contact with it. So water vapor absorbing sunlight isn't a big deal to the weather. It is much more important as a greenhouse gas, redirecting the ground's heat radiation back downward, both day and night.
Also, clouds have a bigger impact on sunlight's energy than water vapor... because they reflect sunlight greatly (something verified by being in the sun when a cloud blocks it!)
- It also can be cloudier in some places during the winter... as shown by this graph for the UK (also true for Iran and Czechia):
(Source: Weather Spark)
Fog and stratus are clouds just as much as any others. Now there tends to be more bubbly cumulus in the summer, more deep convection rising through the atmosphere. But again, this isn't robbing the atmosphere of water vapor, it's an indication of a fair amount.
So why is there more diurnal temperature movement during the summer?
I'd suggest it's because there's a larger diurnal heating difference... in other words the sunlight is "hotter" in the summer (due to a more direct angle), and lasts longer. Then most places still cool back down to nearer the dew point at night. The extra moisture in the summer if anything dampens the diurnal range in summer (see comments about the factors).
Whereas thick persistent clouds in places like London during the winter particularly tend to limit the diurnal range by cooling the daytime and warming the nighttime.