There are two things to consider when exporting for the “general world”:
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If the world-person is using a color-managed viewer, you want to embed the profile corresponding to the gamut of the image data so their viewer knows how to do the display transform. Of note here is that, if the whole world were color-managed, you could export/embed with any profile, even ProPhoto, and all would be well.
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Since the whole world isn’t color-managed, you need to pick a profile to export/embed that corresponds to what you think the predominant display will be. Used to be pretty easy, just use sRGB. Nowadays, a significant minority use high-gamut displays, so you need to decide who you’re going to irritate, the high-gamut folk or the stubborn sRGB folk. I don’t have a high-gamut display, so I can’t tell you which looks less-bad, sRGB on a HG display, or AdobeRGB on a sRGB display.
You shouldn’t export with your display profile, as that’s intended for the output side of the color transform and world-people’s viewers may not be able to use it for the input side.
Yes, I do think AdobeRGB is going to be the new least-common denominator export profile before long…