We cannot discuss here how to “fix” or make playable any content released by Harmonix, including free songs. Having said that, if you are adding customs to a console that already has official content, odds are it’s the customs that are causing songs to disappear. If the official content was causing problems, Harmonix would have heard about it and fixed the issue long ago.
I would recommend first making sure you can still see all songs when having just only official content. If so, then you can start using a process of elimination to find out which custom(s) are causing issues. Once you have isolated the culprits, you can message those customs’ authors to make them aware of issues, or you can try fixing the issues yourself. The latter requires having some familiarity with the files that make up a custom, and how to get to them using programs like C3 CON Tools.
You are about to leave Rhythm Gaming World and access the external songs DB. While no song or file sold for commercially available videogames is hosted there and only fan generated content is allowed, Rhythm Gaming World has no direct control over the database and no files are hosted here. Alright, we had to made you aware of this, now you can visit the database!