If anyone figures out, it’ll be them. Meanwhile in NASA’s 66 years of operation, why hadn’t they figured out how to reuse boosters? Why does it cost taxpayers an exorbitantly greater amount of money for them to do far less with?
Okay...
1) NASA might have been going 66 years, but that doesn't mean the technology to create reusable boosters has existed for 66 years.
2) The USA won the space race in the 1980s, everyone stopped being interested, and space flight was little looked at. This persisted to the early 2000s when a) the Space Shuttles were set for decommissioning and b) interest rekindled because China was becoming competition. At which point the US government effectively threw it open to private tender (e.g. SpaceX). So to all intents and purposes, nobody had been seriously funding NASA to develop better space flight for decades, and when the money finally arrived, much of it was sent to the private sector anyway.
3) NASA does have limitations as a government agency. Failure is embarrassing, and for a government agency funded by taxpayer money like NASA it's particularly embarrassing, especially as that reflects directly on the USA and is potentially hard to justify to the public. NASA therefore is likely to be significantly more risk averse, so it can't afford to blow up every second rocket it tests. It also can't afford to recklessly "move fast and break things" because it's extremely awkward for government agencies to flout laws, regulations and standards made by the government, where private corporations can be less bothered.
4) SpaceX and NASA are not competitors: they are collaborators. It's not just that SpaceX gets a lot of its public funding from NASA, but also that it's able to draw on a lot of NASA's expertise as well. One might note here also that - typically with these sorts of situations - public funding often goes more to "blue sky" research than practical, and it's no exception in space flight. Thus it may not be a surprise that SpaceX is doing most of the work for immediate practical space flight, where NASA is working on stuff with less obvious financial returns.