After installing Lightroom v6.0 and checking for updates, the system appeared to install the latest update (v6.8); however, when I look at System Info in the app, it indicates I am still at v6.0. Should it say v6.8?
I need to confirm before I upgrade my Mac OS because of the known issues running Lightroom 6.0 under Sierra.
This means that the update failed. (which is not uncommon)
But it could also mean that your OS is not compatible with LR 6.8. I am not a Mac user myself, but there is an issue with a certain Mac OS (can't remember which one) which only works with LR up to version 6.3 (I think).
But try downloading the 6.8 update from here Install Photoshop Lightroom and install it manually.
If that doesn't work, hopefully a Mac user can assist you further.