As an Italian (not from Genoa though), pesto to me has always been just a pasta sauce, so I was very surprised to see it used on salads and pizzas abroad. But recently (in the last 2-3 years) I’ve seen quite a few Italian pizzerias (not the most traditional, of course) starting to use it as an alternative choice to the tomato sauce, or, sometimes, in place of the olive oil on top. Not sure if it’s a regional variation gone national or an influence from abroad... but if it’s the latter, no Italian would ever admit that :)

Pesto is very rare on pizzas in Italy, as is chicken and many of the other things that are found in the US.

(It's not bad, it's just that if you were looking for tomato-less pizza in Italy, you wouldn't find pesto.)