Interesting post Acharmenid but I don't think that you can compare Rome to the Assyrians in this, specially during the Empire. Pierre Grimal stated that the Roman trade deficit with the East existed since the late Republic but you don't see any moves to stop them or to impose limitations. The main reason was that it was in the interest of the Senators and the Equites was to keep that trade alive.
Now if any Emperor tried to stop it he would place himself in the right spot to ensure that there would be usurpers, so they couldn't do protectionist measures. The only ones that could had imposed that trade were the early Antonine Emperors, but Trajan's expansion towards the Persian Golf was, according to Grimal, an indication that Rome wanted cheaper and faster trade with the East, not limitations, and he points out that the Dacian gold was what stopped the Gold hemorrhage for some time.
Well the Assyrian nobility would have been in favor of importing luxury goods, that's why they were importing textiles. The Assyrian king was constantly fighting rebellions, often more than one per year, of city governors, large landowners, provincial rulers, etc. The rebels were defeated and replaced with loyalists. While this may seem unsustainable, the Neo-Assyrian Empire lasted almost 300 years. The Assyrian land ownership system was somewhat of a combination of the Roman Empire and feudal Europe, with most Assyrians being tenant farmers, and the Assyrian king was seen as having a similar sort of divine mandate as the Roman Emperor in the Dominate period.
Any Roman Emperor seeking to limit exports of gold and silver or imports of luxury goods would probably see a usurper or even a civil war. But it's not impossible to see the emperor surviving said crisis.
Afterwards it was just something very low on the list of priorities and when stability was what the Emperors wanted and trying to oppose the comercial interests of those with power would be very counterproductive.
The Roman Empire's attempts at stability were one big source of its instability--see all the times that the empire was divided to make it more manageable. Such a large empire is inherently unstable, as was the ancient world in general.
Opposing the commercial interests of those with power will put the Emperor himself in danger, but not necessarily the Empire as a whole.
Less gold in the coins, combined with the fact that the Legions were constantly demanding more bonus/raises meant that you had to use a constantly smaller gold reserve to mint more and more coins, which caused hyperinflation. Also there were Emperors that just coined as much coin as they could, so eventually one silver coin probably had almost no silver on it.
That would be solved by keeping the gold and silver in the empire, at least temporarily, of course, there's no guarantee that it wouldn't be reversed later on by an emperor seeking easy popularity with the Senators and landowners.