There's a lot going on here. First, one of the problems with engineering - and CS has it even worse - is that new ideas and methods get added all the time, but the length of an undergraduate degree stays constant. Compounding the problem, since schools choose or are made to teach more and more things all the time as well (including computing, and topics downstream from computing like "digital media"), one way to deal with more and more topics is to sacrifice depth/rigour on some of the old-fashioned ones like math, so universities have to teach more and more things that they used to rely on students knowing from school as well. The discussion last year about California math reforms and the open letter signed by Scott Aaronson et al. which I'm sure was discussed on HN as well, may stand as an example of this (yes there's a "social justice" angle to it, but the roots of the problem are far deeper than that).
Secondly, a related problem is that a lot of "classical" mathematics was born as an answer to the big questions of the day, and the way it's taught is often just the answer, without any context about the question, and less still about why anyone would care about the question. For example, there's basic logic courses that show 0th-order logic is complete and sound w.r.t the usual rules, but don't give any intuition that this links back to the Entscheidungsproblem and Goedel's theorem and fundamental questions about the nature of the world and philosophy - the mathematical community cared about this at the time! It would be like teaching Turing's halting problem just by giving a definition, a proof and perhaps an example without any idea what question Turing (and Church) were trying to answer in the first place, or the implications for proof-checkers and code-verifiers these days (Coq, TLA+ and so on all exist so Turing's negative result doesn't imply that all is lost).
For differential equations in particular, Cauchy, Sturm-Liouville and others mentioned were all answering (or trying to answer) particular questions that the mathematical community at the time cared about, and without this context the subject can indeed seem dry. I agree with G.-C. Rota (the author of the PDF) that these are not necessarily questions that today's engineers care about, and so yes we can mention that uniqueness theorems exist and move on to linear systems with constant coefficients. If you can't find space to teach and motivate the question, then sure, you don't need to teach the answer either.
As for "functions" that have integrals but no graphs, I also don't like avoiding them by mumbling something about different types of integrals and measures, as if these objects were some kind of organs for sexual reproduction that we can't name because Queen Victoria would blush. No, an ordinary function is something with signature A -> B, and a density on A is a quite ordinary function f with signature S(A) -> B where S is a sigma-algebra that is a subset of the powerset of A and f has certain properties. Usually, but not always, evaluating a density at a set with a single element just gives you 0 and is not very helpful. Measure theory is useful because we can work out certain properties of these functions without always falling back to the powerset definition - that is we can pretend they are a kind of function A -> B in those cases where it does something useful and doesn't break anything else, as long as we understand what's really going on. Case in point: a distribution on the reals that is 1/2 the uniform distribution on [0, 1] plus 1/2 probability mass on the point 2. The probability of any measurable set X is 1/2 the intersection of X and [0, 1], plus 1/2 if 2 is an element of X. There is no mysterious delta function whose value at 2 is exactly "one half infinity" or anything like that, it's just a function from sets to sets where one particular set with one element happens to have a nonzero value. Once you can do this, you are allowed to write 1/2 * U_[0,1] + 1/2 * \delta_2 if you want to.