Why is gold important to Amestris' economy?
Quite simply put, this is a world in which almost anything can be made by putting thought and understanding into it aka alchemy. Alchemy was essentially founded when Amestris was founded, and only existed in Xerxes prior to that. Therefore modern alchemy has been around for approximately 400 years by the time the manga/show began.

The technology at this time was also much less advanced than ours. It would have been difficult to produce bank notes or other such fiat currency that would be difficult to reproduce. An alchemist could even produce all of the necessary components in his own home, and it would be much more difficult enforcing the ban on many specific components.

On the other hand, gold is large and heavy. It is also very rare, with only a tiny fraction of the gold on earth available in the crust. Suddenly having a large supply coming through without any regulations down the supply line would be downright suspicious. Manufacturing gold in amounts large enough to disrupt the economy would surely take notice, and thanks to its weight, would surely be much easier to trace.

Of course, this is all assuming that Amestris was designed to withstand the test of time. Really, the homunculi only needed a (relatively) short-term solution that would keep the populace happy until it was time to murder everyone.