Just out of curiosity I wonder how far you got into "QFT Demystified"? You were last seen entering Chapter 2 (about which I have a small issue on the varational equations).
Truth is that QFT is hard. My courses and textbooks introduced it as "Advanced QM": that is the final 2-3 chapters in a QM book, rather than as a separate topic/book itself. That was hard too mind you. So maybe you should look at your best QM book and have a look at its final 2-3 chapters before doing anything else.
A point to bear in mind is that many of the constructs in QFT are "updated/borrowed" versions of things in QM. A good example of this is the creation and annihilation operators. Their mathematical origin comes from a QM analysis of the Simple Harmonic Oscillator in my QM book (non-self adjoint operators a+ and a- moving up and down the eigenenergy ladder, a-0> = 0 etc.). Nothing to do with QFT as yet. Then as we remember from our early Zee quantum fields are meant to be filled with SHO components all waving about. So we have a field of creation and annihilation operators now. So the maths that is introduced really came from QM.
Also for mathematicians can be the question of how much of the maths is really rigorous. The corresponding maths in QM (or Solid State physics or where-ever the maths is borrowed from) tends to be (essentially) completely rigorous, but the "borrowed" version in QFT might leave more questions. Of course in learning this stuff it is useful to have become familiar with the more rigorous bits in other places. A QFT physicist would argue that the non-rigorousness here wasn't too much of an issue, as it's the physics that counts - or indeed that it's the "results" which count. However the rigorousness of QFT deterioriates as we move towards renormalisation and so on. I am guessing that the rigorousness question will bother you as it bothered me when I tried to learn this stuff.
So bits to concentrate on initially:
1. Creation and Annhilation Operators
2. Klein Gordon Equation
3. Dirac equation and its justification and issues
There is some disagreement about the best way to present spinors which will appear. You will have to muddle along with that one until it is vaguely clear: then mathematical alternatives (which might actually be easier to comprehend ultimately) can be presented.
I bet the Griffiths book is interesting though.
I pretty much finished chapter 2 of QFTD and skimmed the rest to see where it was heading. I have not at all abandoned it - there's too much to finish in any short period of time. I tend to work better when I have the key overview concepts in place - the large-scale map of the territory. Then I can make sense of the details. That's really what I presently lack about QFT (as earlier I did about QM, which was "solved" by the OU course).
I agree with all your points and wait with interest to get to a point where the overall framework seems to make sense or whether one is forced to conclude that it's just something kludgy bolted together. My strategy is one of multiple iterations looping through improving math at each stage.
I hear that even renormalisation is getting quite respectable these days!