I think the gap(s) outlined in the original post of this thread are inevitable since there is a limit to how many courses an undergraduate student can cram into his available time before graduating.
Pertinent material for this discussion is the book "All the Mathematics You Missed But Need to Know for Graduate School" by Thomas Garrity. The goal of this book is ambitious, and so naturally it has its shortcomings. It has a number of errors, and one might dispute Garrity's particular selection of topics. But overall I like the book, and would recommend it to any undergraduate who is considering a professional career as a mathematician.
The main body of the book is a series of short chapters, each a very brief introduction to a particular area of mathematics. This can be a useful read even for the mature mathematician who suspects that their mathematical experience may be a bit parochial.
But one of the most useful parts of the book for me, which I read while working on my master's degree, and which I wish I had been exposed to as an undergraduate, is the introductory material that presents some of the broad patterns that are common across all branches of mathematics, and which form an outline for each of the following chapters. For example, he notes that every branch of mathematics is the study of some particular set of mathematical objects. This study includes questions such as how to tell when two objects of some class are essentially the same (isomorphic); when one is a sub-object of another; how new objects can be constructed from old ones; a notion of maps or morphisms between objects of a class that preserve the essential properties that the objects are supposed to capture; the notion of quotients; etc.
As "muad" noted above, some teachers plan on students learning these principles more or less by induction, from having many concrete examples presented to them, and never mentioning them explicitly. While that may be an ideal way to learn the principles, there is no guarantee that a given student will learn them, regardless of the number of examples given. And I have met some mathematicians who, as far as I can tell, aren't aware of them. As an undergraduate, I came away with the sense that the various branches of mathematics were rather disjointed, with no real common patterns. Blame it on poor instruction, or more likely, my being dense. I think I would have benefitted a lot from someone pointing out these patterns to me in an explicit way.