I am looking for textbooks, lecture notes, lecture videos on a rigorous introduction to the standard model of elementary particles. I'd prefer to not be referred to monographs for an introduction as they tend to be quite dense in my experience. I would however, be welcome to try any that are actually written from a pedagogical perspectove. What I am looking for are resources with a level of rigor equivalent to Thirring's Quantum Mathematical Physics, but I could settle for books like Talagrand's What Is A Quantum Field Theory? which is quite rigorous to some level and sympathetic to concerns for rigor in any non-rigorous essential methods/tools used by physicists.
As for my background, I'm a master's student in physics and I have a fair idea about the physics lingo on these subjects and have taken courses on them. I also have background on axiomatic quantum theory, group theory and differential geometry. However, I'm looking to start afresh with more precise terminology and a well-organised Definition-Theorem-Proof kind of approach. I prefer resources that are comprehensive (it may assume QFT background at the level of Talagrand's aforementioned book) and pedagogically oriented with sufficient attention to the applications (in at least, the form of problems) as opposed to dense travesties that only focus on the mathematics itself. To cite an example for what I mean by this, I would, for learning mathematical analysis choose a Zorich over a Rudin anyday.
I hope that paints a clear picture of what I need and hopefully I can be directed to such resources. I have carefully looked at similar questions to mine but haven't found any material that I would consider introductory, be it in the comments or the answers to those questions. I cannot stress this enough, but I am not looking for mathematical expositions on quantum field theory from scratch.
What I need are only one or a few that build on available knowledge at the level of rigour and content of say, Talagrand's book or even Folland's book to describe the standard model. My main concern is the presentation style besides mathematical rigour and I am not trying to investigate mathematical issues related to the standard model, but only learn about the physics from a mathematical presentation.
PS: I've been recommended Hamilton's Mathematical Gauge Theory before and I quite like it as well, so I'm looking for something similar. However, I'd perhaps benefit from a different type of source that goes into more physical details and intuition as well.
I can extend my suggestion of Hamilton's book from Physics SE a little bit if you want.
So, first of all, as you probably know, at the non-perturbative level, there is, of course, no mathematical rigorous treatment of the standard model, as this is still an open problem (e.g. There is no rigorous construction (yet) known for non-perturbative quantum Yang-Mills theory in 4d (c.f. also Yang-Mills mass gap problem), which is an essential part of the standard model, although there has been some recent progress in lower dimensions within the framework of stochastic quantization). There are also reasons to believe that non-perturbatively, the standard model might not exist at all since it might suffer from triviality problems. (It has recently be shown that this is for example the case for $\Phi^{4}$ theory in 4d, which is used to model the Higgs field in the standard model.)
Secondly, if you talk about "mathematical introduction to the standard model", one also has to point out that there are different mathematical approaches to QFT, such as algebraic QFT, functorial QFT, stochastic quantization, non-commutative geometry, etc., and hence, it also depends a little bit in which particular "approach" you are interested. (Btw, if you want to have a general overview about various approaches and aspects of modern mathematical QFT, I can recommend the article A Perspective on Constructive Quantum Field Theory by S. J. Summers).
From your question, I guess you would like to have literature which is most closely to what physicists do, but in a mathematical language, so similar to Folland's book, which basically is a "bottom-up" approach to mathematical QFT, since he essentially tries to write down the same equations as physicists, but with a higher level of rigour.
Now, as I have already said, the book by Hamilton
is really nice. In this book, the whole Lagrangian of the standard model is described as completely rigorous in the language of differential geometry. Furthermore, the author makes a bridge to the notation used by physicists and discusses many extensions, like neutrino masses, grand unification and supersymmetry. However, the problem of quantization is not part of this discussion. But anyway, if you like these type of books, you can also look into similar books on mathematical gauge theory. One of them, in particular, is
A rigorous treatment of perturbative quantum field theory, via "Causal perturbation theory" a la the Epstein-Glaser approach, containing QED, QCD and in fact also perturbative quantum gravity can be found in the two books of G. Scharf:
Another nice book, more in the non-perturbative direction and in the direction of axiomatic QFT (a la Wightman-Garding, Osterwalder-Schrader) is the textbook
Note that the same author has actually also written a mathematical book about symmetry breaking and the Goldstone theorem, which is also a fundamental part of the standard model (The book is called "Symmetry Breaking", Springer if I recall correctly).
The ingredients needed for the standard model can also be done via "perturbative algebraic QFT". I am not sure if there is any explicit literature about the standard model in this language, but a good starting point for that is the book
Next, as already mentioned above, the standard model also has a very nice formulation in the language of non-commutative geometry. A very nice book, which starts at the very basics of non-commutative geometry and which is discussing the standard model explicitly in some details, is
Last but not least, you might also be interested in the review
on grand unified theories, which also contains a fair amount of details on the standard model.