Washed out colors on After Effects when rendering - Using macOS High Sierra

sRGB is not a valid profile for H.264 video. Internally, the frames are not stored as RGB values, but as YUV. In theory YUV data covers the full range of sRGB, but there limits on extremes. Rec.709 (the standard for HD footage in H.264) only uses the 16-235 "video" range for brightness, so if a playback application is expecting 0-255 "data" levels it will get it completely wrong. There's also rec.2020, which is used for high dynamic range video and cinema. Unless you know what you're doing, avoid it - hardly any consumer displays can handle it properly, neither can playback apps.

Unless you're exporting footage with luminosity outside the broadcast range (e.g. pixels at 255/255/255 white or 0/0/0 black), the exported footage should look visually close to the original when viewed in a color-managed display (i.e. re-imported back into After Effects). You'll sometimes notice the whites are a bit dull, and pixel-peeping will reveal some very minor shifts in hue cause by the RGB<>YUV conversions, but it's not a major problem for video playback on consumer devices where the range of badly-adjusted brightness and contrast settings your audience will have is many times more significant. If you re-import the footage and it looks completely different, and you've checked that it is being read correctly (footage item > interpret footage > color) then there's something fundamentally wrong with your workflow, and we'll need to look at your settings in detail - the project color window, the export options window and the headers of the exported file.

Judging whether the video is "right" using QuickTime Player is just pointless. It is broken, always has been, always will be. Professionals don't touch it with a mucky stick. For desktop playback tests, try VLC (it understands display color profiles and if you change the video output mode to OpenGL it should exactly match the preview windows in AE). If you want a "known good" playback tester separate to After Effects and don't want to install VLC, then import your video into Premiere Pro. If your file works in VLC, then tell the client to use VLC. If they insist on using broken software, they get broken results. What would your mechanic say if you walked in and demanded "I refuse to put oil in this car, but you have to make it work!"

You say this video is destined for a 'digital wall', but unless the wall hardware is calibrated (and it's usually not) then what you actually see on the final screen will be anyone's guess. If a client wants perfect color matching then you have to work with the wall provider to create an end-to-end managed workflow, where they calibrate each panel with a spectrometer and specify a LUT or output transform to use as a virtual "preview filter" when editing. It's not as simple as throwing them an MP4 file and assuming that your RGB 909090 text will appear on the wall as RGB 909090. I guarantee it won't.

If you want to convince clients why "exact color" is a pointless demand when you're producing video, try playing them the same file in different browsers (IE,Chrome,Safari,Firefox) and on YouTube vs Vimeo. Then put it on a USB stick and play it on a TV. Pretty much every instance will look different, but the general public aren't complaining. They just don't notice. If people could tell that their TVs and monitors were all whacked, everyone would be buying calibration pucks. They're not. Although rare these days, walk into any store selling TVs and try to find two that look the same. The salesdroid will tell you the bright blue over-saturated colors are a "special movie enhanced mode" and charge you more for it!