Since software can be copyrighted, like any original work, you will need to respect Fair Use. And even though your use will be educational, a form of entertainment or non-profit, those uses are not exceptions to Fair Use https://en.wikipedia.org/wiki/Fair_use#U.S._fair_use_factors . YouTube also clearly points this out: https://www.youtube.com/yt/copyright/fair-use.html#yt-copyright-protection.
As you point out, "readable source is not equal to open source." So you will need to show only small parts of someone else's website, and not use the capabilities of Developer Tools to modify on the fly the code (technically making a derivative work).
You are free to build your own sample website to use in your videos. It's not that big of a deal to make simple html pages, and you will own the copyright on those and can use them in your videos. You can adapt general ideas and code from other sites for your own demo site, but you can't fully copy or go beyond Fair Use.
But one possible way to do what you want to do (as you originally alluded to by mentioning open source) without coding your own original website is use GPL software as the basis for your code demos in your videos. https://www.gnu.org/licenses/quick-guide-gplv3 . GPL licenses were developed and are advocated by the Free Software Foundation http://www.fsf.org and they show that educational use of software - both in an administrative use in a school and as course materials - is covered in the license:
...(with) Free Software, the teacher can explain the basic subject and
then hand out the source code for the student to read and learn....
Teachers can hand out to students copies of the programs they use in
the classroom so that they can use them at home. With Free Software,
copying is not only authorized, it is encouraged. https://www.gnu.org/education/edu-why.html
One of several web development environments that use GPL (see https://en.wikipedia.org/wiki/List_of_content_management_systems for more) is WordPress: https://wordpress.org/about/license/ WordPress also provides GPL themes - the design and "front end" to a web site - that install with WordPress and that you will typically use as code for your Developer Tools demonstrations.
As per the GPL license, you can modify a WordPress theme or the core libraries on the fly with Developer Tools to use for your educational videos. Read the whole GPL license to find out what you need to do if you want to distribute the modified code to viewers of the videos, if you do that; you need to include a copy of the GPL license.
The easiest thing to may be to go to https://wordpress.com and open a free account and use their hosting for a site to use as a demo in you videos.
I don't see an issue with YouTube and your videos; there are many videos about WordPress on YouTube, including on how to use Developer Tools: https://www.youtube.com/results?search_query=developer+tools . And German law appears to recognize the GPL licenses for software, but I couldn't find anything definitive; one reference is http://www.twobirds.com/en/news/articles/2007/review-german-gnu-general-public-license .