Well, you should use whatever rocks your boat. Personally, I think its a little weird that Twitter Web consumes a lot of CPU as compared to the Desktop app. As for GMail, I find switching between labels (on the web interface) very slow, if you have a ton of labels/emails. So instead of setting up Gears or something, I just started using a native app.
ChromeDeck is pretty great. It's an HTML5 TweetDeck app in the Chrome Web Store that feels almost native. Multicolumn layout for your lists, inline video / photos, and desktop notifications with sound.
https://chrome.google.com/webstore/detail/hbdpomandigafcibbm...
What native app do you use? I use Apple Mail since IMO it has the best IMAP support of the lot, but archiving and filing under labels is a pain. Apple's way is to use drag and drop, which doesn't rock my boat.
Mailplane rocks. It's more of an SSB than a pure native app like Twitter for Mac, but it has a lot of nice little integration points with Mac OS X.
It's not cheap, but it's well worth the price (its also due for a paid upgrade soon, IMO, as I have not paid anything beyond the first version three years ago).
That's an unfortunate side-effect of the Mac app store. It's too bad, because functional demos was one of the nice things about the Mac software ecosystem.
Give it a few days, that was literally only released last night, and they're still waiting for the ad supported version to be approved. Up until then it was in open beta.