We're handling that mostly by increasing the age of retirement, not increasing labor costs.

And immigration. The US would naturally be in population decline without that. Immigration is a two edged sword. On one hand, it removes jobs from existing citizens. On the other hand it funds social security (among other things) and negates problems with population decline.

Who's handling the healthcare costs of the middle-aged workers not old enough to retire?

They are in terms of NI (social security taxes) increases

Ok, so instead of money coming from the retirement system it comes from NI (social security taxes).

How does it result in a drop in healthcare costs while the general population is aging? If a person has health issues at 68, and the retirement age has been just moved to 70, that person won't magically get a clean bill of health.