If part of your workflow involves checking the appearance of sRGB images though a non-color-managed application, you can easily get around this quirk by having a cheap non-wide-gamut monitor sitting next to your expensive wide-gamut monitor and use them in a dual-monitor setup. You can then easily move an image over to the cheap monitor to see how it will look to others on the web. Cheapo monitors are pretty much arbitrarily cheap compared with the good wide-gamut monitor.
Photoshop has always had the option to soft proof to sRGB on your display outside an ICC aware application (Customize Proof setup: Monitor/Windows/Mac RGB).
What the differences here are is the number of visible steps and their differences colorimetrically. Both displays should produce 16.7 million colors but the distances between them is different. For example, R12/G79/B129 and R12/G80/B129 have a greater colorimetric difference (deltaE) on the wider gamut display. So you get a wider gamut true, but you get less granularity between subtle colors you may wish to see.