Is 16GB worth it for a 6-year-old gaming PC?

Yes, the price-to-performance ratio is excellent. An 8GB to 16GB jump is one of the cheapest upgrades you can do short of swapping in an SSD. For a six-year-old PC, used DDR3 or early DDR4 sticks go for $20-30, and the difference in modern games is night and day. The catch is if your CPU is a low-end model like an i3 or FX-4xxx, or your GPU is a GTX 960.