I remember reading the book "Flight" a couple years ago by Chris Kraft. He described what happened the first time they attempted to launch a Mercury capsule on an Atlas rocket. The launch appeared to be going fine, but not long after it entered a cloud bank, things went to hell and they had to destroy the rocket. The Air Force blamed NASA because they'd been launching Atlas missiles for some time, but with no camera footage, all they could do was point fingers at one another. Ultimately, it was determined that the Atlas was weaker than the Air Force admitted, causing it to buckle around Max Q with a Mercury capsule. The rocket was strengthened and the failure didn't happen again.

The point is that the limitation isn't about launching in fog or clouds per se, but about the need to track the launch visually to determine if something was going wrong or to aid in an accident investigation. Imagine how difficult it would've been to determine the cause of the Columbia accident had they not had the footage of the foam striking the Shuttle's wing. There was no apparent telemetry indication of a problem during launch, so all they would've been able to do is analyze the physical wreckage and reentry telemetry to determine the cause. They could've determined that the heat shielding on the left wing failed but they might not have had a clue as to why. Likewise, the camera footage was very valuable in determinine the cause of the Challenger accident.

The Russians don't have to worry about launching in fog. They have a proven launch escape system to protect the crew and the wreckage comes down on land, making an accident investigation much easier. Given the Soyuz booster's excellent track record, they may not even bother with much of an investigation. Back in the late 1980s, I remember reading a report about a Soviet launch failure. Less than 2 weeks later, they launched the replacement satellite. They couldn't have done much of an investigation in so short a time, so I guess they figured that the odds were against two failures in a row.

As for the Shuttle itself, it's an inefficient and expensive launch vehicle. The orbiter itself has a loaded weight of over 200,000 pounds and a few million pounds of propellants are consumed to put about 50,000 pounds into LEO. Most of that propellant mass is needed to lift the orbiter itself, which from a launch perspective is little more than dead weight. All of the processing between launches means the Shuttle costs several hundred million dollars per flight, although the exact cost is a matter of intense debate. All of the labor to refurbish the orbiter for another mission pretty well negates any benefit from reusability.

On the question of payload to different orbital inclinations, keep in mind those numbers of approximately 250,000 pounds to orbit when you count the orbiter and the payload. If increasing the inclination from 28 to 51 degrees meant a 5% reduction in what you could launch to the higher inclination (and that's a number from the anal data store), then your lift to orbit would be reduced by 12,500 pounds. Since, as you pointed out, the weight of the orbiter can't be reduced, that 12,500 pounds comes out of your payload capacity leaving only 37,500 pounds to orbit.

According to this source, the penalty is even greater than 5%.

Launch data is: continuing.
LEO Payload: 24,400 kg. to: 204 km Orbit. at: 28.5 degrees.
Payload: 12,500 kg. to a: space station orbit, 407 km, 51.6 deg inclination trajectory.

Part of the difference is due to the higher inclination and part is due to the higher altitude. Still, you're seeing a reduction in payload of almost 50% to go to the ISS.

By way of comparison, if an expendable booster was capable of launching 50,000 pounds to LEO at 28 degrees inclination and suffered the same 5% reduction going to 51 degrees, the reduction in payload would only be 2,500 pounds. There isn't the huge dead weight issue of the orbiter to consider.