I've not usually seen local Fox channel games listed the day before on Fox Sports Go.
Anyway, I've found this
http://ts.data.foxsports.com/fsdms-static/manual/fsdms_faq
Looks like you CAN watch Fox nfl games on there, but there are device restrictions as it says.
Also to answer that question, yes CBS is taking over Thursday Night Football for the first 8 weeks, they did say that those games would be simulcasted on the NFL Network as well. So far they've only locked into a 1-year contract on this with the potential to expand this if the ratings go as expected.
Your link says this:
"NFL games broadcast on FOX Sports are available in FOX Sports GO on desktop web and tablets. Due to league restrictions, NFL games are not available in FOX Sports GO on phones.*
*You need to receive these channels in your TV channel line-up in order to watch them on FOX Sports GO. Note that some channels may not yet be available for every TV Provider or on every platform."
This is interesting because there are no NFL games broadcast on either Fox Sports 1 or Fox Sports 2 in my TV channel lineup for this coming Sunday. In fact, I've never seen an NFL game broadcast on the Fox Sports 1 or 2 channels. And what I see on Fox Sports Go is mirroring Fox Sports 1. So, I continue to see no evidence Fox Sports Go will give me anything I don't already get on my TV, while it is not even showing the two Fox games I will be getting on my TV on the Fox flagship channel.
The Fox Sports Go schedule shows I'll be treated to a monster truck pull and a Germany vs. Scotland soccer match while Fox NFL games are on my TV.
Unless somebody can produce any evidence to the contrary, Fox Sports Go is nothing more than a way to watch the Fox Sports Channel(s) programming on a portable device or a computer, which seem to broadcast a lot of soccer, UFC, junk sports, etc.