Probably the biggest barrier to space elevators is exactly what SpaceX and Blue Origin are doing - safe, reliable, and eventually (relatively) inexpensive rockets will make the large fixed costs of setting up a space elevator uneconomical.
Once we have a rocket-powered transport network between, say, Earth-orbiting space stations, a moon colony, some near-Earth asteroid mines, fuel depots at lagrange points, a Phobos outpost and robotic factories on Mars, the economic payoff of a space elevator will be more immediate. Plus we could try them out on less massive bodies first, like the moon.
However orbital vertical tethers are doable with Zylon. A series of 3 orbital tethers could move stuff between LEO and the lunar neighborhood. http://hopsblog-hop.blogspot.com/2016/08/tran-cislunar-railr...
Both Musk and Bezos seem well on their way to reusable boosters. But reusable upper stages is another story.
An upper stage re-enters at around 8 km/s. An ~7 km/s delta V budget plus the need to carry a payload means a very slim dry mass fraction, an egg shell space craft.
Recovering and reusing upper stages will be much harder than reusing boosters.
If an elevator could provide more 9s of reliability, that could be huge.
And when we're talking about human spaceflight, consider the work SpaceX and Blue are putting into in-flight abort and propulsive landing - SpaceX's Crew Dragon should be much safer than the Space Shuttle, for example.
Propulsive landing helps for some failure modes but not all. Like I don't think it would have helped the CRS-7 failure had that been manned.
https://www.nasaspaceflight.com/2015/07/saving-spaceship-dra...