Can't be certain, but I know that from a subjective perspective my brain feels the same way browsing timeline as it does when I'm doing math or reasoning intently. The experience of cognitive strain is present in both cases.
One example in the book is that even hard to read fonts can cause cognitive strain and make you judge the content of the text differently. I see that as being largely analogous.
From my reading of the book (and the exerpt you have mentioned here), I'd agree that System 2 is in play when looking at the Timeline; it does create some cognitive load.
However, I can't remember anything backing your assertion that System 2 is more susceptible to advertising. In fact, my reading is the opposite: System 1 cannot help but read words or look at images, and it is easily swayed by various advertising techniques. It would be System 2 that processes what's happening, realizes you are trying to be sold something that you probably don't want/need, then rejects it. Isn't it a lack of engagement of System 2 that leads people to instinctively click ads or follow spam links?
One example in the book is that even hard to read fonts can cause cognitive strain and make you judge the content of the text differently. I see that as being largely analogous.