After some googling I think I found the answer myself. I'll share
"Self-timer will wait and after the delay behave as if you pressed the shutter button. That means locking in focus and exposure, as well as stopping down the aperture if lower than 5.6 or so. This is a mechanical operation and may cause vibration, especially with the FTZ adapter and non-E lenses (i.e. with mechanical aperture lever).
Exposure delay just delays the release of the shutter. All the other steps are done when you press the shutter. Since the aperture closes earlier, it eliminates vibration from closing down the aperture."
So exposure delay is handy with the silent shutter since it will reduce shake caused by closing down the aperture.
If you're getting camera shake because of the lens' aperture closing down, you need to be using a more sturdy tripod.
I've shot mirrorless for years now, and have never had an issue with soft images due to the aperture closing down with using the electronic shutter with a shutter delay or shutter release cable. By all means, continue to use exposure delay, but just know that in practical, real-world situations, there is no difference between the two.
Of course, this would be extremely simple to confirm for yourself as well in about 60 seconds. Simple set up a tripod and camera, and take one image with exposure delay, one image with timer delay, and compare the two.