Indeed, there does seem to be a lot of misconception about what NASA is and does. A lot of things NASA does not do are attributed to it, and a lot of things it does do are ignored.
A reminder - it is the National Aeronautics an Space Administration. One thing people forget is the Aeronautics part. NASA is responsible for a massive amount of fundamental work that directly contributes to advancing the state of the art of ordinary planes that fly in the air.
After the Challenger accident NASA was removed from the commercial launch business. It does not launch any commercial or military satellites. And as has been mentioned above, the vast majority of launch vehicles were not designed by NASA, and were designed and built initially for the purpose of delivering nuclear weapons across the planet. Mercury and Gemini missions were launched on such ICBM vehicles, not on NASA designed vehicles. Some parts of the Saturn vehicles were designed by Von Braun’s team, which was transferred from the US Army to NASA for the express purpose of supporting the Apollo missions. Since the end of Apollo the only launch vehicle NASA has designed has been the shuttle, and arguably the stillborn Aries 1. All the other launches have been purchased by NASA from commercial suppliers that build vehicles for the military and commercial launches. And the vast majority of the design and engineering of the Saturn and Shuttle systems were done in house by the contractors. One would argue that the creation additional of expertise in the contractors from these efforts has been very useful, if not directly quantifiable.
NASA has a critical role is supporting and nurturing these industries. Sometimes there may be altogether a bit much political pork involved in this support, but that is the nature of politics and the manner in which parish pump funding works in the USA. However the role is clear.
The nature of space programs is such that there is both highly specialised engineering involved - so specialised that it is hard to see a direct consumer level spin off. But also an inherent risk averseness and thus conservatism in engineering. Spending 10 million dollars on R&D to maybe find a cheaper technique to replace a 1 million dollar process or design, with one that costs a tenth as much, may be seen as a waste of time and money. Especially if it brings new risks, both technical or in time. OTOH new companies like Space-X are perhaps showing that it is now overdue to revisit this. And NASA’s role is to nurture companies like Space-X. It is a bit like the manner in which the postal service was used to nurture the nascent airline industry.
The usual spinoffs cited (non-stick maps, Tang) are both silly and not true anyway. Spinoffs that matter are really contributions to very high technology areas that we see as beneficiaries of, not direct consumers of. Pushing launch vehicle designs beyond the size needed for ICBM systems probably enabled the introduction of the very large payloads in orbit we see now earlier than if it had been left to commercial interests alone. OTOH you can go too far. The Europeans went too far with Ariane 5 which is now considered too big, and is only commercially viable because it can loft two customers in one go. Its successor will actually be a smaller vehicle.
The other one to be clear of is the division of responsibility between NASA and JPL. JPL has been the prime contractor responsible for a huge number (perhaps more like the vast majority) of probes and landers in interplanetary exploration. Again, usually with NASA as the customer. There can be entire missions that NASA commissions, but does no actual engineering or bending of metal on. Atlas/Centaur pretty much replaced the role of the Shuttle in planetary probes. Witness last night’s launch of Curiosity.