They had several cases along these lines involving several agencies, and I feel like people don’t understand the underlying legal idea - rule making power belongs to Congress. Federal agencies under the executive branch that have rule making powers receive those powers by Congress delegating it to them in a limited fashion through legislation.

Nitpick: rule making power does belong to executive agencies (at least until this SCOTUS decides to reverse Chevron deference). Law-making power resides solely with Congress.

What this means, as you suggest, is that Congress sets up statutory bounds within law, then the responsible executive agencies create rules interpreting them and defining how they'll be enforced. Where cases like this one go wrong is when the agency oversteps the bounds of the law as passed by Congress. At that point, the agency has engaged in creating new law rather than rules, which is why the courts swat them down.

I agree with your overall gist, just feel that's an important distinction to understand the situation.

While many in the US have one or more gun, I would argue most were unaware of bump stocks until the 2017 Las Vegas Shooting. TL;DR, a bump stock uses the recoil of the rifle as a “spring” to help pull the trigger over and over again - effectively behaving like a “machine gun.” As already stated, it does not meet the US legal definition of a machine gun because you’re still firing one round per trigger pull. The bump stock basically makes the trigger pull automatic.

As for why it’s a big debate, read the wiki article. One guy killed A LOT of people pretty quickly using this device to greatly increase his rate of fire. It was a public eye opener for much of the country. So much so, that even Donald f’ing Trump came to the realization that something should be don. He didn’t even really get much push back on it from the right or the NRA. That’s how sobering the massacre was. That said, it happened long enough ago that it’s memory probably isn’t powerful enough for anyone to change the law to ban them and similar mechanisms (see: the binary trigger elsewhere in the comments). If they tried banning them today, the NRA and conservatives would fight it tooth and nail.

That excellent quote of the text you provided spells out that any modifications to a gun that allows any more than a single shot is to be prohibited.

Incorrect.

It prohibits any conversion to a machine gun. The previous sentence has just defined a machine gun. The "by a single function of the trigger" language is what's critical to this case and you're completely ignoring it. When reading laws, you use words however they're explicitly defined if a definition is provided, not how you think they should be defined or would be used in common speech.

Like I said, Gatling guns are pretty highly analogous. They produce what most people would consider automatic fire. They've also consistently been ruled to not meet the definition of a machine gun going back to at least the 1950s because they don't meet that single function of the trigger requirement.

The solution is to change the text of the law.