I worked in meteorology in the 1980s for several years, and one thing I learned was that data is in short supply. I learned that the ultimate truth comes from sending up radiosonde balloons with humidity, wind and pressure sensors. The goal is not just to get data from the ground (there's lots of that kind of data, I'm sure). You have to sample the whole atmosphere to set up the numerical models. You say that about two hundred balloons are sent up every day, and assuming this is done every 6 hours in the continental USA, a back of the envelope calculation says the sample points are about 400 km apart. So the best data for seeding the calculations has very low spatial resolution, much lower than the 2-mile resolution used in the numerical model.
Of course, you can get estimates for water vapour from IR satellite measurements. I saw this done also in the 1980s. At the time I didn't understand all the math used to do this, but remember that it involved taking IR emissions over several wavelength bands, and somehow combining these to infer the water vapour content at various heights in the atmosphere, under each pixel. These satellites certainly have 2-mile spatial resolution, but the problem I see there is that the polar-orbiting satellites that provide this information pass over any spot on earth about 4 times per day, so the temporal resolution is as low as the balloons'.
Finally, data from airlines is going to be largely restricted to heights at cruising altitude, so you're missing a large cross section of the atmosphere there.
And don't get me started on weather observations over the ocean, where there are very few ground stations or balloons.
The issue is that the Navier- Stokes equations being solved in weather forecasting are very sensitive to initial conditions, so it's really crucial to get the data right to set up the calculations. Sitting in my armchair, I remain a bit skeptical that we will ever be able to get the true initial conditions.
Despite all this I'm always impressed that the NWS manages to get pretty decent one-week forecasts out, despite the impossible task they face. There must be some deep voodoo in those numerical models!