The world's output of scientific papers increased exponentially from 1700 to 1950.
One online source is this article (which is concerned with what has happened since then). The author displays a graph (whose source is a 1961 book entitled "Science since Babylon" by Derek da Solla Price) showing exponential increase in the cumulative number of scientific journals founded; an increase by a factor of 10 every 50 years or so, with around 10 journals recorded in 1750.
Perhaps someone can locate similar statistics specific to mathematics, but it's reasonable to expect the same pattern. If so, it is a long time since any individual could follow the primary mathematical literature in anything close to its entirety.
But then, gobbling papers is not how leading mathematicians (or scientists) actually operate.
By making judicious choices of what to pursue when, and with sufficient brilliance and vision, it is possible even today to make decisive contributions to many fields. Serre has done so in, and between, algebraic topology, complex analytic geometry, algebraic geometry, commutative algebra and group theory, and continues to do so in algebraic number theory/representation theory/modular forms.
I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.
I think that Paul Siegel's suggestion is sensible. If you enjoy reading about different parts of math, then build in some time to your schedule for doing this. Especially if you feel that your work on your thesis problem is going nowhere, it can be good to take a break, and putting your problem aside to do some general reading is one way of doing that.
But one thing to bear in mind is that (despite the way it may appear) most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful), so that the problem turns into something you can really solve. This usually takes time, not time reading texts, but time bashing your head against the question. One reason I mention this is that you probably
have more knowledge of the math you will need to solve your question than you think; the difficulty is figuring out how to apply that knowledge, which is something that comes with practice. (Ben Webster's advice along these lines is very good.)
One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.
Best Answer
It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann of Math. 152 (2000), 551-592.
A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. AMS, 132 (2004), 11, 3349--3355.
A. Eremenko and A. Gabrielov, The space of Schwarz--Klein spherical triangles, Journal of Mathematical Physics, Analysis and Geometry, 16, 3 (2020) 263-282.
As you see, they are all published in mainstream math journals. All contain some new results on spherical triangles. And I am not the only person who is involved in this business:
Feng Luo, A characterization of spherical polyhedral surfaces, J. Differential Geom. 74(3): 407-424.
Edit. To address one comment: here is a forthcoming conference on spherical geometry