As for my problem, here was basically the core of it.
The Logic has a mode called "Solo Protect" and it basically disables a track from being put into "solo." I won't explain why this is important, but for example, if there is an effect send coming out of the channel like the one on the left, you'd often not mute it. Perhaps you may be using it to do some side-chain effect controls or triggers. When this mode is in effect there is a red slash (/) over the solo button on the channel strip.
Logic will automatically figure when to Solo Protect a track, but if you are unaware of this feature then you'd be caught for a big surprise thinking that there could be a bug or installation issues with Logic.
At any rate, if you do insist that you would want to turn this off, then all you need to is to press Control while pressing the S button on the strip. Do that again to turn it back on.