For me, the answer is "because native mail clients are so much snappier and responsive".
As well as being able to choose my client based on the interface I prefer, rather than whatever UI the webmail provider decides to give me. And my mail client is also always open. And it uses less memory than a typical browser tab.
The reality I think is, the average user probably doesn't care about snappyness as long as it is not too bothersome. The path of least resistance is opening the webmail (no setup required).
Fair. I just don't always have a browser open. I don't need it when I'm coding and such. If I need to look something up I'll make a note and come back to it. Breaks my flow to just stop and look something up when I can just stub something out and fix it proper later.
As well as being able to choose my client based on the interface I prefer, rather than whatever UI the webmail provider decides to give me. And my mail client is also always open. And it uses less memory than a typical browser tab.