For all we know the reason for the continued lack of a port/re-releases of the original game could be a lot more complex than a dispute over who owns the rights to some of the tracks, but that's pretty much all we have to go on at the moment. The Sonic community has done their best to try and find out what the fuck happened.
I think the alledged lawsuit is supposed to have taken place around 2011 - 2013.
MJ not wanting to be credited because he didn't like the Mega-Drive soundfont and/or SEGA distancing themselves from MJ due to the child-abuse allegations. This left SEGA with an incomplete soundtrack, having to hire additional artists to compose the remainder of the music and having to decide what to do with some of the music already composed by MJ (or Brooks and Buxer). I'm guessing SEGA probably felt they had every right to use the music already composed by MJ who had distanced himself from the project-didn't want to be credited, so they felt they owned it and had every right to include it. However, SEGA wasn't fully aware that other artist such as Buxer (who worked with MJ) & Brooks had contributed to some of the music which made in into the game, and unlike MJ they either didn't want their music to be included or they're not happy with not being credited for their contribution and are claiming royalties. Hence all the legal stuff which has gone down. The Jackson family may also feel they're owed royalties, but I suspect the main issue is with Brooks and Buxer.
When I think of how much content of Sonic 3K was allowed to be included in Mania/Mania Plus with apparently zero repercussions, I believe SEGA do like Sonic 3, are aware the fanbase adore this game and would have released it a few more time since 2011 (along with Sonic 1&2 etc) if it wasn't for a dispute over the ownership of some of the music and the threat of being sued.
Convenient wasn't it that Generations and Sonic Mania both avoided using Carnival Night, Ice Cap or Launch Base (all very popular zones due to their design/music/aesthetic). I wonder if Team Mania had the nerve to suggest re-imagining one of these zones when they were in discussions with SEGA/Iizuka-san as to which Sonic 3 stages to include. If they did have the nerve they probably would have been shot-down so fast, ha.
I do find it strange that the S&K half was included in some of the recent hardware packages (made by ATGames etc) since 2011 (the last time Sonic 3 was rereleased) and now even S&K seems to be off the table-wasn't included in the current-gen SEGA games collection or the Mega-Drive Mni.
Sigh... Well, at least we have Sonic Mania, Sonic 3 Complete, Sonic 3 AIR to keep us busy. Sonic 3 AIR is pretty much the perfect 3K remaster we could ever hope for.