To the point of paying the fee of app in app, you're right that there will always be a penalty. But "will browser apps ever run as fast as native" is not really the right question. The right question is "will browser apps ever be able to run fast enough."
Finally, a clarifying point. When i said "do whatever you do" I meant that a website does the same thing in every browser, not that websites currently can do the same thing as native apps.