The EyeBall Switch won't show up next to my footage.

It's not AV switches you need, It's AV Features. The columns can also be arranged any way you like.

Unfortunately, there is no auto-reset for the timeline layout.

Ok so first you have to check and tell us if your footahe is actually AudioVideo clip and not only Audio clip. If you are sure that you have AV clip (that mean that you can play your video in any video player outside of AE) that mean that AE have some issue with your footrage and it is unable to play video part of it.
I'm saying about case like at that screenshot

first position is where AE see video stream of a clip and second one is a clip without video stream. I\m guessing that your clip show as second one (without eyeball symbol).

That can happen when your footage is compressed with some non standard codec/compression settings or some error in footage that AE can't process. To fix that try to import your footage to Adobe Media Encoder and check if AME see your video stream. If AME can see your video stream of your file - convert it to some intermediatre codec (prores, Adnimation, Cineform etc). After that import your converted file to AE - that should fix the issue.