If by rabbit hole you mean taking the time to know how the system their responsible for maintaining, updating and troubleshooting in production works, I'd say give them as much time as they need. If you think that's taking too long it could be an indicator the system is either too big or too complicated.

I love taking my own time just as much as the next engineer, but if there are extrinsic factors calling for short-term solutions, one can’t take several weeks or months to “research the system” before the crisis is solved.

That's a nice analogy. In a professional setting, spiking on research is time constraint always, and if you start to meandre too much, you will deplete a reputation. But on your own, and with clear goals, as someone here mentioned, by all means dig in as much as you desire if it fulfills you.

On a thread about rabbit holes, it seems safe to suggest that “peddle” was the verb you were looking for.