Thanks to a suggestion from @RonanChardonneau I have developed a browser extension for Chrome and Firefox that makes it really easy to try out how Piwik tracking would work on a website, without needing access to the source by simply adding the tracking code client side.
The extension is open source and the source can be found here: https://github.com/Findus23/piwik-injector
If you run across any issues or have ideas for suggestion, message me or open an issue on the GitHub issue tracker.
And if you want to translate it into your language, join the team on transifex.
I hope this extension helps you as much as it helps @RonanChardonneau, who has made a video showing a use case: