Up till a short time ago Flash was a standard method to build websites. One could argue that Construct (with the canvas element and javascript-->"html5") is quite similar.
And currently many websites out there use the canvas element in combination with javascript to drive parts of their website (or the complete page), replacing Flash based elements. Nothing new under the sun here.
So, yes, you could make a complete website experience in Construct (I believe having read on this forum that some users already did btw) - it even supports responsive elements that automatically adjust to the screen size. In that sense it actually outsmarts Flash. The only real issue would be SEO, because a Construct built web page would be quite opaque to search engines: there is no regular html structure.
Again a comparable situation compared to Flash (which has dreadful SEO)!
It would also be harder to update its content. Much harder - although the Ajax behaviour could be a great help here, and mitigate this issue somewhat. And for a static highly interactive web experience, where SEO is not that important, Construct is a good alternative to pure html and css or Flash.