How urgent is it to intuitively understand Bayesianism?

Maybe just use odds ratios. That's what I use when I'm trying to make updates on the spot.

Not particularly urgent. An understanding of how to update priors (which you can get a good deal of with an intro to stats and probability class) doesn't help dramatically with the real problem of having good priors and correctly evaluating evidence.

What are the practical benefits of having an intuitive understanding of Bayes' Theorem? If it helps, please name an example of how it impacted your day today

I work in tech support (pretty advanced, i.e. I'm routinely dragged into conference calls on 5 minutes notice with 10 people in panic mode because some database cluster is down). Here's a standard situation: "All queries are slow. There are some errors in the log saying something about packets dropped.". So, do I go and investigate all network cards on these 50 machines to see if the firmware is up to date, or do I look for something else? I see people picking the first option all the time. There are error messages, so we have evidence, and that must be it, right? But I have prior knowledge: it's almost never the damn network, so I just ignore that outright, and only come back to it if more plausible causes can be excluded.

Bayes gives me a formal assurance that I'm right to reason this way. I don't really need it quantitatively - just repeating "Base rate fallacy, base rate fallacy" to myself gets me in the right direction - but it's nice to know that there's an exact justification for what I'm doing. Another way would be to learn tons of little heuristics ("No. It's not a compiler bug.", "No. There's not a mistake in this statewide math exam you're taking"), but it's great to look at the underlying principle.

Basic probability is useful to understand, and basic numeracy is useful. Not necessarily on an intuitive level, but on a level where you can back-of-the-envelope or do rough approximations if they are useful (chance of rain, chance of false positive vs disease, etc.)

B is not very useful to really get into unless you are in ML/stats or related area.

I think for most people the limiting factors in "being less crazy" have to do with interpersonal stuff, not math.

First off, I should note that I'm still not really sure what 'Bayesianism' means; I'm interpreting it here as "understanding of conditional probabilities as applied to decision-making".

No human can apply Bayesian reasoning exactly, quantitatively and unaided in everyday life. Learning how to approximate it well enough to tell a computer how to use it for you is a (moderately large) research area. From what you've described, I think you have a decent working qualitative understanding of what it implies for everyday decision-making, and if everyday decision-making is your goal I suspect you might be better-served reading up on common cognitive biases (I heartily recommend /Heuristics and Biases/ ed Kahneman and Tversky as a starting point). Learning probability theory in depth is certainly worthwhile, but in terms of practical benefit outside of the field I suspect most people would be better off reading some cognitive science, some introductory stats and most particularly some experimental design.

Wrt your goals, learning probability theory might make you a better programmer (depends what your interests are and where you are on the skill ladder), but it's almost certainly not the most important thing (if you would like more specific advice on this topic, let me know and I'd be happy to elaborate). I have examples similar to dhoe's, but the important bits of the troubleshooting process for me are "base rate fallacy" and "construct falsifiable hypotheses and test them before jumping to conclusions", not any explicit probability calculation.

I very much understand reductionism and the distinction between the map and the territory

How do you know that you understand those concepts? What does it means for you to say that you understand them?

I very much understand reductionism and the distinction between the map and the territory. And I very much understand that probability is in the mind.

Which, combined together, a problem, because Yudkowskys argument that probability is in the mind attempts, fallaciously, to infer a feature of the territory, the existence of complete causal determinism, from a feature of the map, the way humans think about probability.