Carbon is extremely mobile and C14 and C12 have identical chemistry. The ratio of 14 to 12 just depends on how long the carbon took to get where you found it. It was "modern" going in from the atmosphere, but when was that?
At any rate, there's apparently a fairly even noise floor of .24 or so percent of modern carbon in some old rocks. Evidently, the contamination takes time to get into poorly permeable coals. Above that noise floor, performance is unaffected. So it looks like you can date reliably back to about 50 or 60K and then you're down in the noise no matter how good your instrument is. (That's assuming the same noise floor seen in the coals is present in other materials, but I don't have any actual idea.)
Other elements are less mobile than carbon. There are known problems with K-Ar dating (excess argon, for instance), but there are known work-arounds, such as Ar-Ar. A number of techniques allow Isochron Dating. The point is that there's an array of dating techniques for any suspected age range. Most often they will return the same results for a given sample. (There's no reason why that should be so if creationist criticisms on radiometric techniques are correct, and yet that is so.)
So they are both mobile. And as you pointed out the calibrations of carbon with tree rings have indicated variations of C14 levels in the atmosphere over time.
"There are known problems with K-Ar dating (excess argon, for instance), but there are known work-arounds, such as Ar-Ar. A number of techniques allow Isochron Dating."
Read here for problems with Ar-Ar and Isochron dating methods.
How accurate are radioactive dating methods
Most often they will return the same results for a given sample.
The fact is that we don't know how often the dating methods disagree. To many dating results are ruled out and never published because they disagree with the prevailing world view. In fact the article above talks about how bias is built into the system. The labs even ask for an expected age, before they run the tests. You said there were 40 something methods. Yeah, I bet they do find two or three that will coincide with practically every specimen. But unless they published all results even the ones believed to be errant, there is no way to know how prevalent lack of correlation is among the dating methods.