Data?

In a trial there are two separate jobs. The judge determines what the law is. The "finder of fact" (frequently a jury but sometimes a judge) determines what factually happened. To protect against a judge's mistake, appellate courts review the judges determinations of what "the law" is without any deference. In contrast, factual determinations typically cannot be changed by an appellate court without an overwhelmingly good reason. Therefore, in order to get the law interpreted in the way you wish, you typically need to convince a judge and possibly an appellate panel and possibly a supreme court. While it is true that the law only means what the courts say it means, convincing judges up the scale is easier said than done.

You almost have it, but some details are incorrect. First, only some of the amendments have been "incorporated" against the states through the Fourteenth Amendment. However, most of the really important ones have (e.g. 1st, 4th). Second, if a state court determines a question of federal law and the state appeals process is exhausted, the appeal goes to the federal Supreme Court not the circuit courts.