I love the combination of pesto and chicken. Is there a reason traditional Italian cuisine seems to shy away from it?
I'm assuming you mean basil pesto, although according to an Italian dictionary a pesto is a sauce made by crushing ingredients (pestare is Italian for crushing). Some specific types (like the Genovese one) are protected by law and you can't call something with that denomination unless they have some specific requirements.

As Italian, I've seen basil pesto also as a sandwich sauce together with cold cuts and/or cheese, pizza topping or as lasagna filling as alternative to the ragù (aka Bolognese sauce outside Italy:) I think this answers your first question.

Together with meat, I've never seen it; probably because it can be too strong to cover the meat's flavor. However, a similar preparation with parsley (salsa verde) but probably not as strong usually accompanies the bollito (boiled meat). I'd say the pattern is that we're speaking about not-so-flavorful meats, but here I'm afraid to cross the line and step into speaking about my personal taste.