I would like to add a slightly more technical answer to supplement the other (although it is a bit late). It may be helpful, I think, and it touches a broader issue which interesting. It's a conflict of history of teaching versus history of the mathematical prerequisites for the material.
First of all, supposing the topic material is taught early on, how would that work? I like the following books:---they benefit from the long experience of the authors in teaching this material.
Cartier P, DeWitt-Morette C, 2006, Functional Integration, Cambridge University Press. (B)
DeWitt B, 2003, Global Approach to Quantum Field Theory, I, II, Oxford University Press. (C)
Prior this, the following textbook would introduce all basics, for example to student starting with basic knowledge of calculus.
Choquet-Bruhat Y, DeWitt-Morette C, 1982, Analysis, Manifolds, and Physics, I, Elsevier. (A)
This would be a selfcontained series of courses: A, B, C, which would also teach a good chunk of other material, so a pair of introductory mathematical courses, whose content is covered in (A) would be dropped. So, only one more course, in total, would be required. Why isn't this happening?
(X) The answer is indeed historical, due to historically the prerequisite mathematics for the subject, when first developed, gaining the reputation of being "very advanced material" with few introductory texts, and this reputation stuck. Even when no longer justified, it influences how departments construct their curriculum. Mathematics is synergistic. What was once difficult becomes easier when other concepts are discovered and developed. Advances in method over time made the calculations easier and no longer really something difficult in the usual instances, and plenty of introductory books to the material now exist. However, reputations change more slowly than the environment that generated them.
For example, A.E. above brought up the point that analytic continuation is not taught to undergraduates in many places, and complex variables aren't mandatory. That's my experience too. I suggest the same circumstances being the reason.
For rigorous work with functional integration, distribution methods must be used. (Now I prefer sheaf cohomology methods leading to generalized functions. But the justification and application is the same in this case.) The historical issue is: category theory, sheaves, and generalized functions became widely known and fully developed in the 1970's. (After Feynman's integral was in use for over twenty years!) To know how to do calculations in the arbitrary case, beyond the intuitive setting up of the problems (which was developed 1920's-1950's), these later methods are required.
There are plenty of basic introductory books, published 1990's-present, which are not too advanced for undergraduates. E.g., Lawvere's book, MacLane's several books, etc., assume the reader begins with no knowledge whatsoever about even the most elementary mathematical concepts! And the introductory books on generalized functions and distributions begin easily: right hand limits of appropriate series of sums and products applied to 19th century problems, to solve them more easily than is traditional.
The issue is that they are, in fact, books. Buying a book is not sufficient to learn anything. It must be read. There often (for various institutional reasons) simply isn't enough time in the major curriculum to present all the material without having the undergraduate read something in some book that is not presented in class. But in most universities, at this level, use books, if at all, mostly for homework problems... :_( Historically, the education system has been drifting away from required reading the STEM fields.
On the other hand, the other methods can be presented without required reading. And while the typical undergraduate, coming from a typical high school can read books, they usually don't. If they buy them, it's because they contain required homework problems. Many undergraduate courses are not structured to require reading at all. The textbooks are selected only for their changing homework sets.
As you can imagine, and as is indeed the case, whatever cannot be taught this way, or has the reputation of "advanced" isn't mandatory or taught at all until well into graduate coursework. This is merely one case, it seems to me.