Wow, Diana, this message explains much that I did not previously know and provides a substantial solution to my problem.
I did not know that in basic AirPlay, Apple TV was connecting to and streaming from the server. I always imagined that the video was being transmitted via WiFi from the iPad to Apple TV. This explains a lot about how Apple TV achieves such excellent quality in what seemed such an indirect and potentially problematic signal path.
I also did not know that there were multiple ways to initiate AirPlay. I was only aware of one - the mirroring option. I begin by enabling mirroring, and my iPad screen is immediately visible on the TV. If I begin playing video in what I would call a "well-behaved" application, e.g. Amazon Video, it switches into basic AirPlay automatically. It is this path that worked for me in iOS 10 and that I used frequently. And it is this path that consistently times out in 2:50 for me in iOS 11.
I don't think it's intuitive that connecting AirPlay via the music player, the second of the three methods you explained, would have any impact on video. However, with your encouragement, I discovered and tried that method. Remarkably, it works for video. No more 2:50 timeout. It doesn't mirror, and the mirroring was kind of annoying anyway, but in most cases, including the ones I was most concerned about, Amazon being the prime example, it automatically changes to basic AirPlay and (as I now understand) streams video from the server perfectly with no 2:50 interruption.
This does not seem logical to me, but it does solve my problem and make it unnecessary for me to do what I was right on the verge of doing, which is downgrading back to iOS 10.
There are certain differences in behavior between the two methods that I would like to describe FYI. I will call the mirroring approach Method 1 (M1) and the music approach Method 2 (M2).
In M1, Amazon on the iPad is more than a fixed control screen. It provides the sometimes interesting X-Ray information that Amazon makes available, which tells, for example, what actors are currently on screen. With M2, this is not available; it's a fixed screen. There is a trade-off here. I like the X-Ray info and sometimes refer to it. But it's no contest. M2 is far preferable because it eliminates the crazy-annoying interruption every 3 minutes with its laborious recovery. Amazon may be the only application to which this appiies.
M2 generally works better than M1. There are applications and situations where M1 does NOT switch to basic AirPlay, but continues mirroring the full screen video on the iPad. It does this remarkably well. But it's not nearly as good as basic AirPlay video. Examples of this are the NBC app and some websites on which I watch videos. With both of these, using M1, I was always watching a full screen mirroring on the TV, which is NOT QUITE full screen. With M2, in both of these apps and in other places, I get the control screen only on the iPad and I get true full screen on the TV. It's great! It's how I always thought it should be.
Picking up on a few other points: No VPN in the mix. It's an old Apple TV (maybe 3 years, the little one, but not the new one with apps or the new new one with 4K). My whole reason for getting it was AirPlay, having become a both a Mac user and an iPad user. My iPad lock timeout is set to never, and my Apple TV sleep timeout is set to 1 hour. So, no insights there. And the public WiFi I mentioned has nothing to do with my WiFi router, which is hard wired to my cable modem. At home, I don't see the public WiFi at all. I was just talking about the situation in certain places out and about that I tend to frequent.
I haven't tried using the AirPlay icon in the video application. I'll experiment with that too. I think there may still be some bugs in iOS 11 behind the 2:50 timeout with M1, because it used to work. But what's most important for me is that you've given me a viable way to make this work well enough to again enjoy viewing Amazon Video and NBC and others on my TV via the iPad. For this, I THANK YOU again and again.
