First of all, the Air is indeed lighter than your 13 (165 vs. 174 gr.) while - more importantly - having a much bigger display.
Have you felt a 9 gram weight? It's miniscule. That's less than a 6% reduction in weight from the iPhone 13, not notable.
The screen is not 'much bigger', it's only 0.4 inches. Pull out a ruler and look at how large half an inch is, this is less of a change, and less than 7% larger, that's not 'much bigger'. For practical purposes it's not beneficial.
And secondly, have you held an Air yet? It feels a lot lighter still than the numbers would suggest because the weight is distributed over a larger area. It is a very pronounced psychological effect, same as with the latest, super-thin iPads Pro.
No, not held one yet. It might be a good psychological effect with regards to dynamic handling (moment of inertia), but try holding one statically for five minutes while watching a video or reading a long article and see if that's still a notable difference. I am very skeptical it will be notable.
Now give us an Air that is 150g, or less, and we'll start talking.
-R