My understanding is, the idea that Microsoft and Sony rushed Sega to finish Sonic 2006 was born out of a statement made by Yojiro Ogawa (the man originally responsible for the Wii port of Sonic 2006 and who decided to instead make Secret Rings) where he refers to the fact that the team was "felt pressured" to get the game out.
He never specifies the source of this "pressure".
The reason why we probably ended up with what we see today, involves a lot of reasons. One is that we did want to launch the title around Christmas, and we had the PS3 launch coming up, but we had to develop for Microsoft's 360 at the same time and the team had an awful lot of pressure on them. It was very hard for the team to try and see how we were going to come out with both versions together with just the one team. It was a big challenge.
http://info.sonicretro.org/Game_Development:Sonic_the_Hedgehog_(2006_game)
They extrapolated from his phrasing that this meant Sega was completely innocent and it was Microsoft and Sony pushing for the game to be released before it was finished. In defense of this crazy theory, Sonic 2006
is featured on first-run 360 game console boxes in the "enjoy many great titles like..." section.
But given the game was not really an exclusive for either platform and Microsoft was likely too busy actually pushing Epic to get Gears of War out, I sincerely doubt they gave a shit about Sonic 2006 beyond the notion that "this will probably sell at lot for Sega and we need a kids game to put on the back of our box so we seem diverse".