1/ As guest in current browsers acting as hosts, for the zero-install ubiquity of the Web. This would require WebGPU, Wasm, fast JS<=>Wasm calls, and installable PWAs at least. Gatekeepers notably Apple can and will hinder. Once established, this new hosted app could be nativized to drop the browser dependency.
2/ As new obligate not facultative browser-like app in new device, e.g., XR eyewear. The device would have to take off at very large scale, and its maker(s) would want to do things not easily done via (1). The extant Web would be mapped onto in-world surfaces but the new device and its content stack would support things that go way beyond evolutionary improvements to the Web.
I used to think (2) might happen with Oculus being acquired by Meta, or by Apple eyewear, but I'm less optimistic now. The Big Tech players, especially Apple, can build nice h/w, but it won't get to scale quickly enough to subsume the Web.