Even though our Beit Midrash publishes a weekly dvar torah bulletin, which we make as interest-grabbing as we can with a range of special features, nonetheless, I am against learning Torah, reading parshah sheets and otherwise engaging in study during davening. During the breaks between aliyot, perhaps [for what is kriat haTorah if not communal Torah study], but not during davening.
This is not about halachic issues related to learning Torah during the repetition of the amidah; my point is spiritual, not technical, and it's about the entirety of davening, from Modeh Ani to Adon Olam.
The other day, I found the following metaphor to explain this to my children:
Imagine that you are a chef, and you prepare dough for pizza, and you slather on tomato sauce.
Then you decide that it would be wonderful to have chocolate chip cookies, and so you add chocolate chips.
Then, reverting to pizza-making mode, you put on cheese, but then you add the vanilla needed for cookies, and you mix it all together.
Then you cut it into small balls and put it on trays in the oven.
The result: Chocolate Chip Pizza. And while I love pizza, and I love chocolate chip cookies, I'm not having any chocolate chip pizza.
The same is true during davening. Davening to HaShem is great. Learning Torah is great. But mixing them together is self-destructive; the davening lacks emotional commitment, and the learning lacks focus and concentration. Neither is successful; the result is trash, a waste of time and resources.
I admit that I will look to learn something if the chazan is dragging on for so long that I cannot focus on davening anyway. Absent that, though, learning plus davening equals Chocolate Chip Pizza, and I'm not interested.