The FAA almost certainly accepted this risk because it's just a mission hazard not a safety hazard and they probably didn't have the budget for it anymore. They'll never say more than "we handled it well", because that just invites the cycle of stupidity that makes people less safe.
But that team had kind of an "unfair" advantage in that were able to program in assembly code on bare metal with no real software stack. Whereas the rest of us are forced to build on a foundation of sand using multiple layers of low-quality third-party software in order to deliver any useful functionality.
The Space Shuttle’s avionics software was not written in assembly, rather HAL/S, a high-level language invented for the project. Assembly was mainly used for the custom real-time OS kernel. They also maintained their HAL/S toolchain, which was written in XPL-a PL/I dialect which was popular for compiler development in the 1970s. The development environment ran on IBM mainframes, and the main CPUs on the Shuttles were the aerospace derivatives of the IBM S/360 mainframe architecture, System/4pi, model AP-101. The same CPUs were used by USAF (e.g. the B-1 bomber), but USAF mainly used JOVIAL to program theirs. Another big user of JOVIAL was the FAA, who used it to write a lot of their original mainframe-based air traffic control software (FAA HOST).
The Space Shuttle team inventing their own programming language was a byproduct of the time the project started (1970s). If they’d started a decade later, they probably would have used Ada instead. But Ada didn’t exist yet, and they thought inventing their own language was a better choice than JOVIAL
Which also suggests to me that the problem isn't per-se the glacial progress & approval processes, but that they're using the wrong glacial processes.
If this FAA software was developed with something like the Space Shuttle's process it would still take forever to change something, but at least you'd end up with a function that you could mathematically prove would be able to handle any conceivable input.
I think you're never going to convince the government of a "move fast" culture. Even if the overall cost of grounding planes would be less than the cost of more reliability they'd never go for it.
Too many people's asses are on the line, and they're not having to spend their own money. The only thing they have to "pay" for is possible loss of face, or loss of political capital, both of which can be insured against effectively for free with taxpayer dollars.
But you might just be able to convince them that they're using the wrong sort of bureaucracy. You'd still spend a billion or two on something that should cost a million, but at least you'd get actual reliability as a result.