There are advantages and disadvantages to both - web apps are available anywhere you can find an Internet connection and browsers connected to the Internet are in practice more readily available than your device filled with native apps.
I don't know about you, but the same phone travels in my pocket, so those apps are available to me 99% of the time. The available everywhere advantage of the webapp was a big deal because computers were heavy, and that's been diminished.
That said, some sort of standard interoperability between phone and laptop/desktop apps would be nice, and iCloud APIs seem to be a stab at that, but it's probably a pipe dream to hope that that will become universal. So maybe you're right.
You can certainly read emails on it, however if you want to write emails that are longer than a couple of words, nothing beats the 100 WPM a real keyboard gives you. And most things you do in your daily work are not feasible on a phone.
And sure, my laptop always accompanies me, however there are always emergencies and on my laptop I prefer web interfaces anyway. I haven't used email clients on my laptop ever since I discovered GMail.
