It is an impressive thing to watch seasoned studio musicians "do their thing".
Also nice that they recorded a session to share...

One thing that they did not show in this case is how good those guys are at writing charts.
They had one done beforehand - most likely by Paul since he was obviously in charge.
In a typical demo session, they will be doing 4-5 songs for a "customer" (songwriter(s)) who simply bring a cassette tape of a quick guitar/vocal - as rudimentary as it gets.
They slap that in a player and the musicians write the chart as they listen. One time through. They then chat around about what they hear to make sure they all have they same thing. Then, as you see, a quick "groove check", some discussion about various parts and hit record. One take is typical (once all the preliminary set up in done). A few punch ins for a missed note here or there. This is typical of songwriter demo sessions. Generally, there is a long standing relationship between the writers and the musicians, so everyone knows what they are after.

Generally, one of the songwriters is in the vocal booth to provide a scratch vocal as they go.
A seasoned demo singer will be brought in later to do the vocals - in a separate session - you can't waste the musicians time getting the vocals right. It makes for some funny vocals when a a couple of guys have written a "girl song"...

Keep in mind what this costs. A 4-5 song session (which is typical at high end studios because everyone gets more bang for their buck) will cost $2000 ($400-$500 per song). So BIAB cost basically the same as one song - and it's FREE after that!

Also...keep in mind, these guys do this 24/7. And they only have to do their part. And they work together day-in and day-out. Coming up with the grooves and licks that fit what they hear on that simple cassette demo is nothing to them - it's automatic.
As you do your BIAB production, you have to do all the thinking. Think like a guitar player. Think like a fiddle player. A steel player. A drummer (who can do that?!?)
SO, it is going to be a bit tougher to "get that groove". But the tracks are there that will give it to you.
You just need to learn to listen like a producer - and then poke around RTs until you find something that will give you what you need.