I will try to make this as less subjective: Till what point do you continue doing exercises at end of the chapter to learn something: I do it in Grad school and that's where I really learn. Do professors/post docs learning about new fields do it too?
To reiterate and "second" other answers... but to try to re-frame the sense of the question, also, ... I'd agree that one should engage with ideas, but not necessarily on the terms dictated by textbooks. As alluded to in other answers, one of the benefits of experience is knowing which exercises might be worth doing... as opposed to being unable to tell, or merely being compelled to do a high volume of exercises toward a grade.
Memories fade with time, even for the best memories, and unused ideas fade out of one's mind. On one hand, this can be an indicator that a highly-touted thing has no genuine utility, at least in one's own professional life. On another hand, sometimes the pay-off occurs so much later than one's introduction to an idea, perhaps decades earlier, that the relevant connections are very dim. To guard against "fatal dim-ness", occasional "refreshing" of memories is necessary. No, this is not quite the same as "doing exercises", in the important sense that, at this sort of "later" point in one's life, it often happens that reviewing larger points is sufficient to revivify specific memories about relevant (as opposed to traditional-pedantic) details.
One not-entirely-realistic attitude, but quite useful as an ideal of sorts, is to consider the notion that one changes oneself sufficiently so that the ideas of a subject seem entirely reasonable, simplest-thing-in-the-world... rather than unintuitive and unanticipate-able. That is, the meta-exercise is to change oneself so that no further exercises are necessary (apart from perhaps the "refreshing" of memories).
These comments have ignored the hugely significant psychological point that one should as-soon-as-possible recover from the burden of thinking in terms of approval of some authority figure. To say the least, this is not an intellectually honest or psychologically healthy criterion to apply to one's activities. Pity that "school" engenders this so powerfully. That is, don't embark upon a regimen for the approval of others, but to achieve your own objectives. Oop, yes, that entails trying to understand what your own objectives might be. Yes, if you are an undergrad or grad student, or even a postdoc or junior person, this is complicated. But, I recommend, do recognize that issue: one should be trying to think for oneself, but with incomplete information, or bad information, and lack of experience... ?
Last, "of course", one should not use "doing exercises" as an avoidance mechanism to allow avoiding facing more amorphous, real issues. "Doing exercises" is not the stairway to heaven, any more than a stringent exercise program guarantees professional athletic success, etc.
The relevant summary might be to start thinking in other terms than "exercises". After all, there are no "exercises" that qualify you to buy lunch, or mail a letter... The fixity of novice mathematicians on "school" and its ... mixed... credentialling system is in many ways very noisy and non-informative. Even if one needs to fit into that, one should not think too much in those terms.
Keep doing it as long as it’s useful to you.
I’m a couple years into post-doc’ing, and find exercises just as useful as I did in grad school; and I know professors at multiple stages of their career who appear (from conversation) to still frequently do exercises. On the other hand, I know at least one person who claims to have never particularly found exercises helpful.
So it certainly varies from person to person whether exercises are/remain useful; for a given individual, though, I have no idea how it typically changes over time. For my part, the main change over the last few years is in how selective I can be about exercises: I can now generally look quickly over a page of exercises and gauge pretty consistently which ones will be essentially routine, and which will require some real thought. Both kinds can still be of interest (at least to me), but for different purposes.
Exercises fall at various levels in Bloom's Taxonomy. Exercises at the higher end of the taxonomy are harder to do, but verify more objectives, since they rely on understanding and applying knowledge from the lower levels.
To try and answer your question, I'll generalize a bit and say that academics are by nature quick learners for the lower-level objectives (knowledge, comprehension). Exercises at this level are probably less useful than Upper-level exercises (application, analysis, evaluation, creation).
However, whenever I haven't taught material in a while, re-doing some of the knowledge exercises is a great way to freshen up and be sharp for class.