I’m afraid there are some mistakes in ur workflow:
1. “my monitor is an NEC 3090WQXi and I’m using the Windows ICC profile as supplied by NEC”
You paid a huge amount of money for a good wide gamut monitor but ur saying that u didn’t hardware calibrated the display. First thing tommorow morning, get a i1Display Pro colorimeter or the Nec one and proceed to calibrate. Use BasicColor, I highly recommend it. If I remember correctly this is an old monitor so the colour must have shifted a lot. In any case even for a brand new monitor hardware calibration is mandatory.
2. “When I select this profile, the AE project colors immediately appears a little washed out. OK, no problem; I add an adjustment layer……”
This is totally wrong. I speculate here but u probably created the graphics on the same wide gamut monitor and u tinkered with the colours to look good on it. But u did not colour managed them for sRGB.
Of course when u import them in AE and suddenly u restrict the colour space to sRGB they will wash out.

Watch this graph for a wide gamut monitor. sRGB is a subset of what ur monitor is capable of.
3. “Then I render it (uncompressed AVI) Making sure that that sRGB IEC61966-2.1 is selected in the color management tab as the output color profile. Lo and behold, the resultant video looks completely wrong, with around 15% more contrast and over-saturated”
Well, this is totaly expected !! Welcome to the world of video players which don’t have a clue about colour management but are displaying video on a wide gamut lcd. :))
On a serious note, ur video player doesn’t colour manage the video and this is very problematic on a wide gamut lcd.
The solution: delete & forget 🙂 all the video players u have installed in ur system and get MPC-HC. This is the only video player I know of that does colour management.
Look at this
My setup: Dell 2711 wide gamut profiled with i1Display Pro & BasicColor
In the image you have same frame from a MOV file originating from a 5D Mark III. (BTW, now 5D Mark III uses Rec. 709 as colour matrix).
AE is colour managed using as a workspace profile Rec.709.
MPC-HC is colour managed.Under colour maangement in MPC-HC u can specify the input type and I presume that the player assumes Rec. 709 for HDTV type. (sRGB and Rec. 709 have the same primaries and white point, btw)
VLC (or another player in ur case) is not colour managed, hence the shift in colour.
A good thing to know is that the graphic card can do very strange and undocumented things to the YUV to RGB conversion. The trick is to download the K-Lite Codec Pack (MPC-HC is included) and bypass the grapic card for this conversion.

Regards,
Cristian