First, although that YouTube warehouse collapse was scary, I don't think you need to worry about that level of failure. They obviously overloaded the shelving, probably by a lot and just a small amount of damage to a, just ONE vertical set it all off.
The first thing that jumps out to me is your choice of putting the mounting point for the swivel axis of the casters INBOARD of the legs. That is a big problem because, with the wheel swiveled by normal movement of the rack, the base is decreased on the leading edge of that motion. This makes it a LOT easier for it to topple over if even a small bump is encountered. The weight of all that equipment high above the floor provides a lot of momentum up there and when the base stops, the top will not even hesitate. And OVER SHE GOES. If you draw the lever arms you will see that as it starts to tip over, the front vertical leg will actually get LOWER and the weight itself will add to the instability.
You need to have the wheels OUTSIDE of the base of the rack or, at their worst possible position, under the outer edges of the base. This means that your horizontal members (angle or channel) needs to extend at least by the radius of the castor wheels PLUS half the width of the plate of the castors that mounts to that horizontal member. It should also be that far outside of the left and right sides, but the rack is probably wider than it is deep so you have a bit more fudge room there.
And those horizontal members should be either welded to the legs or UNDER the bottoms of those legs. If you insist on bolting them to the sides, use at least grade 5 bolts, not the common hardware store grade 3 ones. Grade 8 would be much better.
The casters: Swivel casters are a bad choice. I would use fixed casters and bolt them on at angles that allow the rack to swing in an arc if that is needed. Fixed casters will always contact the floor at the same point relative to the rack so they can be mounted on shorter horizontal members. This would be a LOT safer.
The rack itself: The photo does not show the entire rack, but I will bet that there is very little cross bracing on the sides and NONE on the front and rear. It probably relies on the strength of the rather small joints as seen at that rather wimpy cross piece in the photo and the small, 45 deg gusset plate that is partially visible at the top edge of the photo. Those aren't much. I would add some 45 degree X style cross braces from the front to the back legs. And also the same at the rear. That will add a great amount of strength and stability to the overall rack. If you use screws to mount the fronts of the equipment in the professional rack-style, put them both at the top and the bottom of those front panels. That will help brace the front. I do not recommend using shelves or trays for the equipment in this, inexpensive style of rack.
After saying that I do not suggest shelves, I will say that one shelf at the bottom and another one at the top will provide cross bracing in the horizontal plane. These could be sheet metal or plywood.