Wind turbines and solar panels are not precision measurement instruments. They are instruments to create electricity. So, the answer in general is no, but there may be some validity in the idea for at least offshore wind (but however, the measurement instruments need to be separately installed to the platforms).
Let's take a look at how weather is measured and predicted.
All the time, a number of geostationary satellites are imaging planet Earth. They are far away at the geostationary orbit at ~36 000 km altitude, so they see the ~6370 km radius Earth at a very small angle of view. Despite this, they have very precise measurement instruments that take 3700 x 3700 pixel visible/infrared photographs from their viewpoint. The 3700 pixels corresponds to twice the Earth radius or 12740 km, so one pixel is about 3.44 km x 3.44 km. Of the 3700 x 3700 pixel image, a fraction of pi/4 is planet Earth, so they have around ~10 million useful pixels. The geostationary satellite images, when analyzed, can reveal information about cloudiness, cloud height, cloud top temperature, cloud type, etc. Far more than just the effective cloudiness level.
As you correctly observed, there is a ground based measurement network of surface measurement stations. True, there are not that many of them, but they measure lots of parameters: wind, temperature, cloudiness, air pressure, humidity, rain, snow, etc. However, they are limited to ground, rain and cloud cover observations. You cannot observe temperature at kilometer height from these ground based measurement stations.
Often times, weather balloons are released. The advantage is that weather balloons rise to the upper atmosphere and during their path, and transmit continuously the current conditions at the current location.
Some airplanes have weather measurement instruments. There are over 40 000 airports in the world. Each airport may have numerous runways, and the paths individual airplanes take can wildly differ, so an airport is a very useful source of weather measurement information, if the airplanes have weather measurement instruments.
There are also weather radars in the ground. They measure precipitation at a very high resolution.
The information obtained from these measurements are entered to a numerical weather prediction model. The NWP model then runs the equations forwards to determine the predicted weather all around planet Earth as a function of time. The usefulness of the NWP models is limited to 7-14 days. Note that the NWP requires an initial information to run the equations forwards, and the initial information needs to contain full atmosphere of the planet Earth. So, if you know the conditions at the surface, but not conditions at the upper atmosphere, you're out of luck. So, airplanes, weather balloons and weather satellites all of which give some information about the upper atmosphere are useful.
Now, what information could you obtain from solar panels? Effective cloudiness mainly, from the power production. Unfortunately, the efficiency of solar panels declines as a function of time, so they are not precision measurement instruments even when calibrated. Also, their power production varies as a function of temperature so without a separate thermometer, you cannot tell the effective cloudiness level because you do not know the temperature. They function only during the day, whereas weather satellites see the infrared channels also during the night. The solar panels would have to compete with cloudiness measurements from numerous weather satellites each of which has ~10 million useful pixels. Will they be tens of millions of solar installations? Most certainly someday, yes. But, unlike the weather satellites, they are not precision instruments and they give data only from populated areas, unlike weather satellites that image the whole planet. I don't think they will win the competition with satellites. (There are also polar orbiting satellites that see the Earth from a closer point of view, giving even more information than what geostationary satellites give.)
About wind turbines, then. They would measure direction and speed of wind. However, they are not precision instruments either. The problem with the 341 000 wind turbines is that they are scattered in wind farms. If one wind farm has 50 turbines, we have 341 000 / 50 = 6820 wind farms. In contrast to these 6820 wind farms, there are over 40 000 airports. A wind turbine can measure only the wind (versus an airplane which gives temperature, pressure, humidity, wind, etc.) and can measure it only at the height of the turbine, ~200 meters probably for state of the art turbines. So, in comparison to an airplane that takes off, flies in the upper atmosphere and lands, from/to >40 000 airports, giving lots of measured information, a wind turbines gives only information at ~200 meters height, only for 6820 wind farms, and only wind and nothing else. I don't think a wind turbine will win the weather measurement competition with airplanes.
About the only validity in the idea of using renewables to improve weather prediction is to install a precision weather measurement instrument to offshore wind farms. The problem with surface measurement stations is that they are not usually located in the sea. (Sometimes, and old ship may be anchored as a sea-based surface measurement station.) An offshore wind farm would offer plenty of possibilities to locate automatic weather measurement stations on the sea. The platform to mount the instrument on is already installed, so the costs would be cheap. But, it would not be the renewable energy production itself measuring the weather, it would be just shared platform with proper weather measurement instruments.
However, the weather measurement instruments mounted on offshore wind farms would not replace weather satellites as the most important source of numerical weather prediction initial data. The difference, if any, in the length of useful forecast days would probably be in the third decimal. A far more useful source of improvement is simply Moore's law which continuously improves the computing power of supercomputers. Due to the chaotic nature of the equations, even a number of years of Moore's law will not allow us to forecast weather a month from now.
Disclaimer: I am employed in the weather service business.
