>Did they really die out, or did the population just merge with modern humans.

Most populations die out. Most sapiens. Most neanderthals. Most everything. The chance of becoming an ancestor is very, very low.

Neanderthals were not slowly observed. Most of our neanderthal DNA comes from a single mixing event/period.

Evolutionary history is a history of bottlenecks. Small populations that survive and become large populations. The rest don't make it.