Why don’t air travel booking sites use the "shopping cart" UI concept?

I am working on an Airline site refactor, and I realized that the abandonment after searching and selecting a flight is very high.

I think for returning visitors it would be nice having a way to continue with the search/booking journey. I am thinking of something like a shopping basket on top right corner, so when you come back to the page, an expandable menu shows you your last started flight booking or so...

The problem is that I couldn't see any airline site with this interaction, and I wonder whether I am missing something.

Thanks!