Does anyone know what exactly, nearly got that game pulled at all?
Unfortunately, the actual issue will likely never be known, at least not any time soon. I will detail several technical reasons for this which I believe are objectively compelling enough to base my answer upon.
The primary issue preventing the diagnosis of that glitch is a fundamental matter of hardware architecture, and how using software to force one architecture to behave another will more often than not be extremely difficult and sometimes completely impossible. This is the reason why emulators in general are difficult to develop, highly difficult to develop well, and darn near impossible to develop so well that they'll run all games for a given platform without issue. This is why there's so many crappy/unfinished/abandoned emulator programs out there and so few fantastic ones.
The second issue is that emulators don't actually emulate the hardware components and chipsets of a console. They just duplicate enough of the high level functions of the components that they're able to play the games. Even the emulators regarded as "perfect" are only cycle-accurate (correct timing), which is plenty for playing games but still not 100% hardware simulation (chip-accurate emulation).
The Donkey Kong glitch had to have been the result of some freak error between the memory management coded into the game and the actual memory control unit of the N64, meaning that to try to debug it via emulation on a PC, we would need chip-accurate emulation to fully simulate each and every logic chip in the N64's ludicrously complex Reality Co-Processor, funky 9-bit memory bus, and every other component. Unfortunately, this is literally impossible with current technology.
Most people will think the idea of a modern PC being unable to simulate the hardware of a 20-year-old Nintendo seems crazy. However, an Intel-architecture PC and an N64 are so vastly different in how they process data that a PC's extra CPU cores, exponentially faster clock, and several-thousand times more RAM isn't even remotely enough computing power to 100% accurately emulate the internal logic of an N64. In fact, a top end PC wouldn't even come close to being able to do chip-accurate emulation for an SNES or even an NES. Intel CPU's just aren't built for it. Even the one cycle-accurate N64 emulator is known to get choppy on anything less than an i7.
Finally, just for funsies let's say those technical limitations magically disappear tomorrow and someone hands us a flawless emulator to use for testing and debugging. I'd still be willing to bet against the true cause of the glitch ever being found. You have to consider the fact that we're talking about Rare in its heyday. This game was programmed by arguably the most talented and competent developers to ever write software for the N64 and who would have been more intimately familiar with the tiniest details of the hardware itself than anyone else in the world aside from the engineers who designed it. They would have known every aspect of the game's memory management logic inside and out because they literally designed and wrote the code for it. For the mystery glitch to elude them so completely as to make Rare say 'screw it' and give away expansion packs, the root cause would have to be something unbelievably obscure and impossible to track down with logical troubleshooting.
If the people who are the world's foremost experts on the DK64 software couldn't fix it when it was literally their job to do so, I doubt anyone else will discover the cause unless they just happen to by total accident.