Web app performance will always be playing catchup with native, and while the performance gap may shrink (perhaps to the point of irrelevance) it will never be eliminated. The OS and device makers are making the rules.
The real future is in our increasing ability to recognize and build to unique platform advantages in integrated ways. Dismissing the web platform as merely 'distribution' minimizes the value of a universally accessible, instant connection to everyone. It allows for different kinds of experiences unrelated to native performance. As the performance gap b/w platforms decreases, we may see more rich experiences for a web audience that's interacting in realtime with native or console users, broadening what's possible for everyone.
Wolf3D is also an interesting example because the rendering is done in software (so your high powered HD video card doesn't help at all) and I guess a lot of development time was spent writing asm routines/hacks to take shortcuts with the necessary math.
To be fair, a lot of the reason that people are remaking these old games is because people like playing old style games and the people building these demos are probably doing it in a very small team (or even solo).
It should certainly be possible to remake Half-Life (maybe even HL2?) entirely in javascript/WebGL or make more sophisticated 3D games. But we won't see it until some bigger gamedev is willing to risk a fair chunk of change on the browser as a gaming platform.
They are, just as they're active on mobile platforms - EA, Epic and Square-Enix are notable examples. But they don't bother with AAA games (much as they like to pretend they are and attach them to those franchises), instead they target the casual player, that spends up to half an hour on a game per time, and/or that gets addicted to addicting games (w/ microtransactions). And the formula works for them; low development cost (5 - 10 dev team), high returns. Epic's Infinity Blade (and sequel) are the highest grossing games (in terms of development cost versus returns) of all time, at least, for Epic.
Most FPS still give me motion sickness, but I distinctly remember throwing up with the first Wolfenstein.
The transformative power of all of those (excepting maybe GMail) was realized in the form of cgi-generated webpages (click, refresh). Performance is not a requirement for those applications, whereas it is very much a requirement for interactive games. 20 years might be a little too harsh, but only a little.
With the exponential rate of development I doubt it'll take a full 20 years to catch up, but it'll still be a while before we've completely reinvented all of the wheels...
But yeah, for the forseeable future, "look at how I re-implemented this ancient technology with modern foundations, and it only took half the time! (And 10% of the efficiency and 100x more RAM)" is all we have to look forward to :-(