Nope, view functions as intended.

The text in the post says: "Below is the code of a rudimentary viewer for a DOM structure of an HTML page". So view shows the "skeleton" of the web page DOM structure, which is occasionally handy. It was never claimed in the post that view would allow one to actually view a web page in a way similar to the web browser experience. This would've been a vastly harder problem to solve in WL, and also rather useless - if one needs to view a web page, one should just use the browser.

The problem with Global` variables is that they are global... :) That is, if you open draw[] twice, it will share state and interfere with each copy.

Even if your GUI will not support multiple instances / prevent its existance, it is good to wrap definitions with private context by using standard BeginPackage + Begin["`Private`"] to not run out of symbols for multiple projects / other people projects.

At the end you could use $CellContext [1] but it is not convenient, out of the box, requires deeper understanding and slightly adapted setup to work so untill you know what is the end shape of your gui, it is not worth for beginners.

1. https://mathematica.stackexchange.com/a/102391/5478