Only if the people viewing the images are dumb enough to use a non color managed browser. Otherwise any working space is game as it is in Photoshop and lots of other app's. And if that dumb person has a wide gamut display and isn't using a color managed browser, sRGB looks awful. People who care about how images appear work with color managed app's so the previews are correct. It's impossible to post anything to the web and expect everyone will see it correctly.
Much of what I've learned about color management over the years, I learned from Andrew. You would do well to read the materials on his website about the subject.
I will just add, in reference to the bolded sentences above, that it's also a good idea to include an embedded profile
tag in all images posted on the web, especially on photography sites like this one, where a substantial number of viewers can be expected to have wide gamut monitors. That's because even some ostensibly color managed browsers mishandle images lacking an embedded profile--for instance, Chrome on a PC. In those cases an image converted to sRGB, but without an embedded sRGB tag, can look greatly over saturated on wide gamut monitors. Whereas they will still look reasonably OK on a standard monitor. That has happened several times on this forum over the last few months, generating arguments among members who see different colors depending on the browser and type of monitor they are using.