The pipe joints are not the problem at your premises.
No wonder the plumbers have responded as indicated.
MUD? Some municipal water supplier? Or a private water company?
Have you had the water tested by a reputable independent laboratory? Local health dept may be able to recommend one.

I am just as confused as the plumbers you have contacted. Rubbing your hand on a pipe in the attic does not tell you anything about what is in the pipe or about the condition of the soldered joints. Also, having the joints resoldered is unlikely to have any impact on improving the water quality.

I would add a good quality whole house water filter on the incoming water line, and give it time to work on cleaning up the incoming water. Then see if you still have water problems in the house. 13 year old copper pipes should flush clean in a short time once the incoming water is clean. You might still have to clean some grit out of faucets, but that should be about all.