One problem, but not the whole problem, is the line
but it's a paid software and I'm looking for a free solution.
Implying that you want a free program because the one that works is not free.
The other problem is that the question is far too broad and there are many website technologies and ways to obfuscate or otherwise bypass the "normal" means of just giving a user a link to a video and letting them download it.
The reason one program works will be because of time invested in making it work.
We cannot go through every website trying to figure out how it obfuscates its download and showing you how to get a link for it. Specific website problems are also off-topic here.
You already know you can go through the source and hopefully extract a link, but your problem is with websites designed to prevent that and also skirts away from "fair use". If they are actively working to prevent you downloading the videos then arguably you don't have a right to download them.
So I chose one of four reasons why I didn't think the question was appropriate.
- Seems to be asking for a product recommendation. In hindsightthis could be the wrong close reason.
- Is too broad, asking how to deal with every website not specific ones.
- Websites are technically off topic here.
- Your question is technically about defeating copyright mechanisms you don't have a right to break. While we allow questions about youtube downloaders, we cannot make assumptions about how you use those tools and only assume you have a right to the content in the first place. Your question isn't about the tool, it is how to break DRM in the same way as the tool and is a grey area as it is us breaking DRM and.