The entire planet was still recovering from the worst recession since WWII, a recession that finally killed off, or critically damaged, most of the watch making companies in Switzerland. None of them had been thriving since the early seventies, under a number of pressures, not least direct competition from Japan and especially Japanese quartz.
Rolex, without shareholders and with vast strategic reserves, due to the unique way Wilsdorf had set them up weathered this storm better than any other high mid market mass manufacture, but even they were not making or selling much.
Also, compared to the then previous market giants like Eterna, Cyma or Omega, Rolex were still fairly small and hadn't yet quite developed the reputation that they went on to have. This was largely by recovering from the recessions and shocks of the seventies and eighties in a position to take advantage of the new found prosperity of the later eighties.
Wilsdorf's original philosophy of producing watches that were aspirationally affordable to anyone fitted the eighties recovery perfectly and that's why availability eventually soared.
Going by serial numbers: in 1985 8,614,000 in 1986 8,900,000. So that's only 286,000 watches produced in that period. I doubt many of them were gold day dates I'm afraid, But birth year watches are usually a bit of a struggle and all the more satisfying for it. You might want to look to sellers in Japan or the Middle East, where, ironically, there might have been more of a market for that sort of luxury! Good luck.