I do not believe that Shibuya station is the busiest in the world. My impression is that honor belongs to nearby Shinjuku station.

Shibuya has two private rail companies (Tokyu and Keio), three subway lines (Ginza, Hanzomon, and Fukutoshin) as well as 4 JR tracks.

Shinjuku has three private rail companies (Keio, Odakyu, and Seibu -- although the last one is not directly connected to the rest), three subway lines (Oedo, Shinjuku, and Maronouchi) as well as 14 JR tracks. I bet that even not counting Seibu, Shinjuku still beats Shibuya.

Shinjuku and Shibuya are both complicated. But Tokyo Sta is the worst.

Well, Shibuya's actually straightforward if you're just getting off JR and then exiting out Hachiko Exit. Mind you, I'm not sure I've gotten off NEx at Shibuya (usually I'm getting off Yamanote or Saikyo for JR to Shibuya) but I presume NEx stops in the same area as Yamanote/Saikyo. So I think the OP will be fine. It's when you have to transfer from JR to other lines (eg, Toyoko/Fukutoshin) that it gets complicated with lots of stairs involved.

Shibuya has two private rail companies (Tokyu and Keio)

Actually three private rail companies - Tokyu, Keio, and JR East.

Shibuya used to be the third-busiest station, but I think it recently caught up with and passed Ikebukuro.

JR East is a private railco?

Yes, Japan National Railway was privatized on April 1, 1987 and JR East has been a private company since.

Shibuya also has Eidan subway! :)

it would be tight but with good planing in advance, do-able.

Unfortunately not.

Map of Shibuya station

I not sure the above map is easy to understand. I have been to Shibuya station so many times over how many decades, so I can easily understand the map, but for the first time visitor...

NEx uses platform 3 & 4. There is big orange arrow connecting map of platform 3 & 4 with the rest of Shibuya station. Hope can see that platform 3 & 4 is located at southern end of platform for Yamanote Line.

When getting off NEx at Shibuya walk the platform toward north. If getting off NEx from Narita Airport then walk toward front of the train. At the northern end of platform 3 & 4 there is an escalator and a stair going up. That lead to the Central Gate of Shibuya Station, which is little bit of walk.

Once out of the Central Gate then go left till you are out side facing the road. Then make right turn toward north will take you to Hachiko exit area, which is the famous Shibuya crossing.

I think 6 hours can be more than enough time for trip to Shibuya, or may not be. Worse situation is this:

Took 30 minutes to go through passport control and customs at NRT, which can happen when there are multiple international arrivals and the line at the passport control is long. Then will be another 5 - 10 minute to get to JR Narita Airport station and buy a ticket.

What if at that moment, NEx has just departed? Then wait another 30 munities for NEx. Which mean can end up spending entire one hour just to leave the airport. Narita Airport - Shibuya is 1 hrs. 20 min. by NEx. This translates to under worse situation can take 2 hrs. 20 minutes just to get to Shibuya.

I do think you really want to be back at NRT security/departure passport control 30 minutes before the departure. Which means departing Shibuya back to Narita Airport least two hours before the flight. This will leave 1 - 1.5 hrs. at Shibuya.

If this is your first time at Shibuya or Tokyo then I will put cushion on schedule since likely you will spend some time figuring around which way to go for what.

Just checked the most recent statistics and yes Shinjuku has Shibuya beat by just a bit. Both handle over one billion people a year lol!

The various parts of JR are "private" in the same sense that NTT and DoCoMo are. They are majority owned by the Japanese government and run so they return maximum revenue to the government without being subject to the scrutiny that a government agency might face (but rarely is in Japan).