There are many situations in which we have a clear collective understanding of intention, or goals, and examples which persuade us that these goals are plausible, as well as illustrating apparent causal mechanisms identifiable as "reasons" for things being the way they are.
For many reasons, the contemporary style of proof-writing, especially in intensely didactic situations, is wildly different from any attempt at "persuasion", or invocation of "intuition". Indeed, on one hand, one certainly wants whatever "physical" intuition is available, but, on the other, one eventually wants a proof that independent of literal physical phenomena or analogies.
The fundamental theorem of calculus, the intermediate value theorem, and many other of the basics of calculus, were visibly more-or-less true long before they were formalizable at all... and their utility gave some interest to eventual formal proofs, even while those formalized arguments, needing also formalized contexts, do not (to my perception) clarify or enhance the physical intuitions.
At least one particular pattern of events produces unhelpful, unpersuasive proofs: situations in which some interesting, important examples are fairly immediate, but it is less clear what "the general case" might be, and how to describe it. For example, the topic of "topological vector spaces" is arguably very practical, insofar as many natural spaces of functions should appear here, but there has been sufficient abstraction and eponymous-adjectification so that statements of theorems, and their proofs, can be couched in terms that do not obviously refer to any example of interest... and it often requires some scholarly chops to figure out the relevance of the iconic theorems to any particular tangible example.
In a different vein, the Cayley-Hamilton theorem? Historically/originally, just proven for the two-by-two case and three-by-three, and in the two-by-two the most naive possible computation succeeds. The three-by-three case can also succeed by most-naive means, but it is ghastly. The general case needs at least one of several more-abstract devices.
Not really undergrad, but certainly discussion of Sobolev spaces and regularity properties on Riemannian manifolds is best introduced by $\mathbb R^n$ and $n$-fold products of circles.
Generalities about automorphic forms and $L$-functions, and "Whittaker models" are arguably best introduced via discussion of the holomorphic case, with explicit exponentials for "Whittaker functions", etc.
Riemann surfaces are probably best introduced via the square root function, logarithm, and so on, prior to "covering by an atlas of charts..."
And from the other side, it is invariably wise to check an idea by considering examples... and if one cannot find any tangible example, or cannot determine the truth or falsity of the immediate question, that in itself is telling.