I think there was a period where you could mostly get away with not using a color-managed workflow if you were producing content for the web. But that’s not true anymore.
Our viewing environments were largely un-managed, and displays were often poorly calibrated. Now, we might have high-quality, wide-gamut, factory-calibrated built-in displays in our laptops, tablets, and phones. Browsers now color-managed; colors outside sRGB can be specified in CSS and images are rendered correctly.
Our phones capture wide-gamut photographs and our painting and drawing apps support wide-gamut colorspaces. Our image content is increasingly not sRGB.
Workflows and pipelines need to understand and correctly preserve or handle color management when publishing to the web, or an increasing number of images will appear incorrectly desaturated or washed-out on users’ displays.