Washed-out colours in After Effects when rendering

There is a ton of things wrong with (broadcast standard) video and color management for desktops that is too complex to explain a simple video editor/animator. But a starting point... sRGB doesn't really exist in video land, and especially in Adobe land everyhting you render (that isn't HDR) is pretty much always encoded with the same profile: Rec.709.
There are nuances with how this is managed throughout a system, mac/windows/vlc/quicktime/webbrowsers etc. Even with all the knowledge you won't be able to get an accurate match if what you've mastered is according to reference spec meaning finishing in a dim lit room to a Rec.709 100nits display. 

But for pure web users you could do a sort of 'hack'. If you rely on your colormanagement in AE being sRGB for sake of bringing non sRGB elements (like an AdobeRGB photo/psd?) into your project you could add an adjustment layer on top of your final comp with a Color Profile Converter that converts Rec.709 to sRGB. For sanity sake, I would also set your project color management settings to Rec.709 instead of sRGB because that is what it should be for video. (your os color profiling will convert this to your display so you won't visually see any difference).

The added adjustment layer will "mess up" your colors but the render should look the same as inside AE without that layer.

Alternatively you don't manage at all at the project level, but that only works if your display is already sRGB and not something wider gamut. For non sRGB assets you'd manually convert them with the Color Profile Converter effect. You'd just view the result as is in AE and when you render it will also still look the same. Then just hope that everyone that will view your content is in the same boat as you viewing wise 🙂 which they probably are.

They still looked washed out if i upload the MP4s to a website so the issue's not exclusive to Quicktime.