I think you need to make distinction between website and web application. The latter came later and is by definition a lot more complex, I assume this is mainly where the premise originates from.
But having gone through building websites with frames, tables, png's for box shadows and border radius, jQuery and php based web apps, and having spend many days making my grunt/gulp/webpack config work and supporting IE, I can with great certainty say things have never been easier than they are now.
Now, is it necessary [to build webapps]; Yes. Usability expectations have gone up, the level of details and polishing has gone up, more interactivity, and more sophisticated visual elements. Webapp now do things native applications could not do back in the day.