As Alpha said, Jon almost certainly owns the rights to anything they recorded, expecially in the last 10-15 years. Some contracts will stipulate that they own anything the band (jon) does while under contract to them, but that's a very bad, very old school contract. Lots of contracts were like that in the 80s/90s (remember credits on other people's records, such a "Jon Bon Jovi appears courtesy of Mercury Records"), which could explain why we are only getting later footage - I expect Jon negotiated a new contract at some point.

Some of the stuff that was recorded for broadcast will be owned by the respective stations/channels.

What Jon doesn't own is all of the publishing to the songs that he co-wrote. He can't simply give these songs away for free without paying the appropraite royalties, which is the only reason that I can think of for only sharing 3 songs per show?