Given your budget I think you are limited to second hand of the age you mention.
Back in the 80's and 90's bikes were generally steel and only in the late 90's did aluminium wheels really become reliable. So most of the surviving bikes from that period are pretty low quality. By asking for a decent one by todays standards you're pushing the original price up, and looking at a racing bike rather than a mass market bike.
Even today, most racing bikes are for adults, and adults who are tall. Back then road bikes tended to be taller than they are today as well - one side effect of the improvement in materials has been the ability to reduce frame sizes without compromising strength. So a small frame by todays standards would have been considered very small 20 years ago.
If I look around right now for a light, fast second hand bike for someone under 160cm tall those are hard to find. Sure, you can buy new ones, but they're usually owned by competitive riders and passed down the family or friendship trees until they are worn out. At cycle clubs they do come up, but there's usually a kind of waiting list formed by parents having a word with whoever owns the bike.
Back when those bikes weren't as cheap and there were fewer of them that effect was more pronounced. Imagine they started at, say, $3000 instead of $1000 new today. That's the effect. So when people did have one, every kid in the club who was at all promising got to ride that bike. By the time it was retired it was pretty conclusively ridden into the ground. If you find it second hand now it'll be ready to be restored for display, not riding.
What I suggest doing is buying a cheap, heavy second hand kids bike while you save and search. It is just patience and putting up want ads. I mean paper ones - print out a few and put them up in bike shops and velodromes. You need to jog someone's memory rather than finding someone who is looking for you. Also, watch for end of season sales at the bigger shops. Bike shops don't usually stock the small sizes because they don't sell. But if there is one, it probably won't have sold, so it'll be in the sale. You might get a new "last season" bike for half price.