As mathematics grows and diversifies beyond belief, surely the collection of topics that every mathematician must know is shrinking fast. One can carry out serious mathematical research in one area while knowing very little of another, even when many mathematicians regard that other area as fundamentally important. Thus, the assumption in the question that there is anything substantial in the list of tppics topics that all ALL mathematicians must know seems to me unwarranted. Of course, the interdiscplinary interdisciplinary work that connects widely separated research areas is often very important (as well as difficult), but a lot of progress is also made within the various specialities without interacting with other areas. But for someone to to insist that every mathematician must know category theory, say, or homology, seems to exhibit just as narrow a conception of mathematics as to insist that every mathematician must know how to program. There have been profound mathematical advances in subjects requiring none of that knowledge. All other things being equal, of course, a mathematician would be better off knowing some category theory or logic or homology or programming, but in practice, all other things are not equal, since we must all choose how best to spend our time, choosing the topics that seem most relevant to the ressearch research we seek to undertake.
Ultimately, we need all kinds of mathematicians: some who are deeply specialized, some who know various areas to build the bridges that can connect diverse subjects, some who know how to communicate ideas from one area to another, and others who know how to communicate the deep ideas of one area to the future specialists in that area, or to the public. Perhaps the intersection of the knowledge of all these people is rather smaller than one might think, and this isn't necessarily a problem.
Contemporary mathematical research is indeed a big tent, as Charlie Frohman said in the comments.
