"Why are browsers accepting invalid HTML in the first place?"
For compatibility reasons, and in the case of newer browsers, because HTML5 dictates an algorithm for parsing even invalid documents.
Earlier HTML specifications were ambiguous on many situations,
such as what happens when the wrong tag is seen, or inconsistent nesting of
tags, such as
But now the HTML5 specification includes a much less ambiguous algorithm for parsing HTML documents. Note that the algorithm includes points where "parse errors" can occur. But these parse errors usually don't stop a modern browser from displaying an HTML document, although the browser is free to display parse errors in its developer tools if it chooses to:
[U]ser agents, while parsing an HTML document, may abort the parser at the first parse error that they encounter for which they do not wish to apply the rules described in this specification. [Emphasis added.]
But again, no modern browser, to my knowledge, aborts parsing a document this early because of parse errors (barring extraordinary situations, such as running out of memory).
On the adsense.google.com situation: This probably has nothing to do with invalid HTML, but rather, perhaps, because IE9 and earlier's DOM support is not sufficient for adsense.google.com's needs.
