Had lunch with a friend of mine I haven't seen in quite a few years. I met him through PCA and we didn't have much in common once I wasn't an owner anymore. He comes from money and turned that money into a LOT more money - but is still a reasonably down-to-earth guy. His car collection is fun and quirky, ranging from an eclectic bunch of 911s to some cool Mercedes including a 300SEL 6.3 that is literally like the day it left the factory.
Last summer he took Euro delivery of a new C4S which he drives year-round unless the weather/roads are really bad, then he's got a handful of SUVs to pick from.
He figured it would be fun to lap the 'Ring while he was there. He's a very competent track driver - one of the faster guys on our track days. He takes it seriously, having attended several high performance driving schools.
I was so jealous! The famous Nurburgring! how cool! Well, maybe not so much. He said...
1. The surface is well taken care of, but narrow. Really narrow. There's just not the room of dedicated race tracks and not being a "properly designed" road course, apexes are all weird and visibility is poor.
2. The range of speeds is truly terrifying. He didn't expect to be a hot shoe in a standard C4S, so his plan was to stay off to the right and let the people in truly fast cars with good knowledge of the route pass. Well, there's no shortage of 2.0L Golfs out for some laps and they're sloooooooow. Thanks to the narrow track, he found passing difficult and ended up spending a lot more time in slow traffic than he'd have wanted to.
3. While he picked his passing spots carefully, he said many DID NOT. The track's lack of passing areas meant guys were very aggressive and risky when passing. At track events in North America (and he's been to lots around Canada and the US) he said a lot of the driver's would be black flagged for the passing that happens without anyone caring on The Ring.
4. People crash. A lot. And the ones that are still on the track have no concept of what a yellow flag means. He was surprised that he managed to get his car home without a lot of body damaged because he would slow for yellow flags - a lot of cars would just barrel up behind and then past him and some ended up wadded up into the existing crash.
5. He tried to plan for what he'd do if he did go off. Stay in the car and use it to protect himself while yellow-flag ignoring maniacs piled into him, or make a mad dash and try to escape? On a North American track, you NEVER get out of the car unless instructed by a marshal unless you're on fire and figure you're about to burn to death. He said there, guys would hop out, and wander around. On the track side of the barriers! 
While I might still be a little envious of his fantastic car collection, I'm not at all jealous of is Nurburgring experience...
I drove 4 laps of the Nurburgring back in the summer of 2011 in my Subaru and although I agree with some of what you say, I still found it a memorable experience.
1) Yes, it is relatively narrow for a race track but it is a proper racetrack and if you know what you're doing, you can overtake slower traffic. It is in fact one of the world's most historic race tracks, as well as being possibly the longest and most technically challenging too. It takes years of hands-on experience to "learn" and master the track rather than just a few laps and driving simulators like Grand Turismo etc don't do the camber and elevation changes justice. It does also still host race meetings such as the Nurburgring 24 Hours, although I don't think Formaula 1 Grand Prix for a number of years.
2) Technically it's a toll road, not a track day. As such, it's less expensive than a track day and is open to the public more days of the year than a normal race track. You are not required to wear a crash helmet and don't have to book, you can just turn up and drive. You also have to pit after every lap and go back through a barrier to go out again, so you can't do a flying lap. If you were able to, as the manufacturers who have testing facilities there can, you would be able to build up some serious speed on the start/finish straight. Because of the nature of how the touristfahrten is run, you're not necessarily going to attract exclusively just the dedicated track day crowd. There will be tourist buses, family wagons/vans and camper vans running around there too. It must be one of the few race tracks in the world where you can rent a track-prepped car and these tend to be "driven like you stole it".
3) While it's true that there aren't many long, wide straights to overtake on, if slower traffic keeps to the right, you can overtake it just fine. It's such a big circuit (over 8 miles) that it's difficult to police like you would a normal size circuit.
4) There are a lot of crashes (one during my four laps) and even a few fatalities each year. The track punishes mistakes and has minimal run-off areas compared to more modern tracks, so if you come into a corner too hot or cook your brakes, then there's a good chance that you're going to hit something (hopefully just the barrier). It's also worth noting that if you crash and they have to stop the session, you will be charged to have your car recovered and fined for lost revenue. Bearing in mind, the size of the track, they are pretty efficient at responding to accidents, warning other drivers, clearing the debris and keeping the session running whenever possible. Another reason for the accidents is that the track is up in the Eiffel mountains where the weather/visibility can change at any moment and it can rain a lot turning the rack from dry and sticky to wet and slippery.
5) Best approach if you have an off is to get out of your vehicle when there is a gap in the traffic and make your way safely to the other side of the barriers and wait for help.
Personally, I wasn't that quick, as I was on my way to Switzerland with my wife at the time and wanted to preserve my brakes, tires and bodywork for the trip. I just wanted to enjoy the experience and maxed out around 120mph on the start/finish straight. I did however manage to find some space and get to use the full width of the track whilst being mindful of faster traffic (a lot of Porsche 911s and BMW M3s) coming from behind. However, there are quite a few locals who drive the circuit and are pretty fast in some quite humble machinery e.g. old Honfa Civics and VW Golfs. If you find some space, you might get in a 10 minute lap in a fastish car but a sub 8-minute lap would be pure fantasy on an open session.
I wouldn't make a special trip just to drive the Nurburgring (although many people do) but as a stop on a European road trip, then yes, definitely. It's not a fast circuit for the inexperienced though. Something like Spa in Belgium would be better for that. The Germans are quite big on health and safety, so I'm kind of surprised that it's still running. I don't know of any other tracks that have such an informal set-up. This has its disadvantages but if you get the chance then I would do it again. Just take it easy, don't race, don't crash and just enjoy the experience of driving one of the world's most famous and technical race tracks.
Many of the top global car manufacturer's have a technical base there for testing, as it's probably the closest a race circuit gets to real roads with different cambers, surface changes, bends and elevation changes etc, so it's a good place to do high speed handling testing and much of the track time is sold to car manufacturers for private testing (and these are many of the YouTube videos that you may have seen). One doesn't think of Germany as a top tourist destination but actually the countryside and roads around the area are pretty nice and it's a lot more affordable than Italy and Switzerland to stay/eat.
