Iirc they did put some more characters from Rise of Lyric into the TV show in the latter half of the first season, like Perci. But besides that, yeah there's not that much tying Rise of Lyric to the show. Sticks also seems to just be inserted into the game as an NPC to sorta tie it into the show.
I am not sure it matters, but it's pretty clear that Big Red Button had a drastically different story and environment in mind than the TV show people did, and the game people lost the creative arguments. Which is a bit of a shame as it looks like from the alpha builds they had some good ideas. It could have been a solid Rayman 2-esque platformer even if we would question it being an actual "Sonic" game.
Here's one hell of a whopper update on the game's development history. Spanish site Sonic Paradise published a whole slew of new information about the game's development; thanks to the efforts of researcher MotoBadnik, who was not only able to uncover some prototype trailers of the game from 2012 and 2013 (back when it was known as "Sonic Synergy"), but also secured an interview with a former Big Red Button employee (who requested anonymity). You can read a translation of the article below, courtesy of Fusion Ellipis from SSMB. You can also download a Big Red Button internal PDF of February 2014 here, it's a design guide in which you can see various concepts and ideas for the game and how it evolved, including several things discarded. Sonic Paradise have also posted more concept art (you can see them both at the site and at Fusion Ellipis's translated post at the SSMB link above; I tried to add some to the post here but the images refused to embed), as well as posted links to the two prototype trailers mentioned above (which I've embedded below).
I'm not sure whether to be furious at the fact that we missed an opportunity for a Chao return, or relieved at the fact that at least they didn't show back up in this game of all things.
Sounds like Boom could have been an even bigger departure than Lost world was if Sega didn't step in.
Anyone remember that old poll Sega did to see what direction the fans would like a future Sonic title to go and one of the options was an origin story? Looks like this is what was originally going to come of it.
There was also the Facebook poll that helped influence Generations' level choices, that one didn't go as badly (depending on who you ask).