The tool for that purpose is normally the use of quotation marks. Large news outlets rarely make up quotes out of whole cloth. That is not just bad praxis, but entirely unnecessary to skew coverage, if they are enforcing a particular perspective for whatever reason.

Look, I do think that large journalistic outlets are a bit stuck on old newpaper composition practices and hyperlinking and multimedia tools are underused. Specifically, news sites tend to be very reluctant to use external links for a number of reasons, including the fact that they want to keep you inside their publication to serve you ads, so external links are not a beneficial business practice.

That said, you are very fixated on a problem that either doesn't exist or doesn't make up the bulk of the issue you're trying to fix. There are plenty of instances where a quote that sounds bad is used out of context, and in most cases the in-context quote is widely available. The outrage machine is fed in opinion pieces, live TV debate and online chatter. There is no need to misquote in an article for that, and there is no evidence of the mitigating value of having a link to a different article. Which, incidentally, may not even be available for free distribution in the first place.