The whole “space age” thing was overblown to begin with. It’s interesting that back in the early 1940s the handful of people seriously interested in rocketry calculated how much payload a chemically fueled rocket could launch into orbit, came up with the correct figure of 1-5% of takeoff mass, and promptly decided no one would ever bother. Up until 1957 most people thought space flight wouldn’t happen until someone could build a atomic-powered superrocket. But the US and USSR developed rockets for use as ICBMs, and advances in electronics made unmanned payloads possible. Arguably it was the transistor, the compact thermonuclear warhead and the maser that were the real innovations behind the “space age”.
For about ten years from 1957 to 1967 many believed that manned spaceflight would have an immediate and vital role in the strategic balance of the Cold War. Literature from the era is full of things liked Air Force bases on the Moon, (manned) orbital reconnissance flights, nukes in orbit, hijacking or destroying enemy satellites, etc., etc. But none of those things turned out to be practical. It didn’t help that outer space proved to be much less hospitable to human life then previously known: the last shred of hope that Mars or Venus could be humanly habitable were dashed, and space proved to be a high-radiation environment (Van Allen belts, solar flares). Ironically, even before Neal Armstrong set foot on the moon, the Apollo program was running on inertia, and since the Soviet lunar booster was a total flop, we no longer even had to keep up with the Joneskis. With limited military or economic potential, space exploration, manned and unmanned, was cut back to what we as a society were actually willing to pay just for scientific knowledge: not all that much.
I would say the space age died somewhere in 1974, a dismal year in multiple respects.