Not really. If you converted to sRGB (as you should), it doesn't really matter if you embed it or not. All and any browser, color managed or not, would display it the same.
I don't think that's correct except for monitors with a colour space equal to sRGB.
Most browsers colour-manage images containing embedded profiles, but they don't colour-manage images that do not contain embedded profiles. That's not the same as assuming sRGB, they simply don't colour manage images that don't have embedded profiles.
If you have a standard-gamut monitor with a colour space close to sRGB, then colour-managing an sRGB image is pretty much a null conversion. As a result, not colour-managing an sRGB image means that it will look approximately the same anyway.
However, if you have a wide-gamut monitor, then sRGB images will be greatly over-saturated if not colour-managed.