I think the relevant sense of urgency would reasonably be about engagement with serious mathematics, not necessarily keeping a pretense that one has "started research" at a quasi-expert level. Nor do I personally see urgency in "passing exams", since that is a typically very-limited enterprise, but might take long enough that one's original interests, enthusiasm, and motivations are forgotten, which would be a bad thing.
And, yes, standard coursework tends to suffer from so much inertia that it is not reliably helpful. But that is not to say that there's little to learn. In principle, sure, one can learn everything needed from papers, books, etc., whether physical or electronic. However, there is so much of this material that some guidance is good. Preferably expert guidance, rather than novices-leading-the-novice.
In particular, in terms of "choosing research problems", while it's good to follow up on one's enthusiasms, one should not be surprised to discover that, very likely, an enthusiasm based on just-a-little information inexpertly chosen is ... naive. Of course! Seemingly natural research problems are all too easily accidentally intractable, or perceived as routine exercises by experts, or of no interest to experts, etc. If one is independently wealthy and just doing math for personal satisfaction, none of this matters. However, if a PhD is in part warm-up to making a living as professional mathematician, the expert-standards do matter, however artifactual they may be.
(The issue of "understanding" one's "project" is ambiguous! One may doubt on general principles that one understands fully, indeed, or suspect that one understands to-some-extent naively. That's entirely reasonable. I would claim that much of the work done to finish a PhD in mathematics these days amounts to getting up to speed on techniques and viewpoints that experts might declare "standard" or "routine"... since, after all, some facility with such methods does sometimes enable one to do things that would seem amazing or unbelievable from a more low-tech viewpoint. After all, that's most of the point of sophisticated methods and ideas.)
So: the urgency is to interact with experts as early as possible, yes.
But, again, it is not accurate to say that this excludes coursework, nor includes it, either. It does not exclude or include following one's own curiosity. It is a separate thread.