Honestly these icons are really nice and are a good medium between clean and defined. I think Material failed as a whole to accomplish the paper look but this is very well done and has subtle but not 2008-era giga-drop-shadow depth. Well done, weather app.

Regarding material design: I do like depth in web pages and when it is used selectively it can look amazing. Actually some of the most well designed websites - personal preference - use depth layers.

Have a look at stripe's site!

However I agree with you that pure Material design can lead to bad results - not bad but really outdated. The whole thing paper on a paper on a paper is just too much.

I'd rather have flat components (flat within) and only use depth to separated cards/buttons/etc. from the background.