It seems that my main issue with sRGB files not being displayed correctly on my Wide Gamut display stems from them being untagged. Each file needs to have an embedded profile to explain what colour space it was intended to be displayed on. This profile is used by the browser (if the browser is colour manage aware) to display the file correctly.
My issue this week stemmed from an error on my part. When I was preparing an image for the web, I failed to tag it with the sRGB profile. Now that I've done that, Safari is now displaying the colours of the file correctly.