I remember having a problem with a specific AT case, but had nothing to do with mounting the board. I had a slim desktop style case, which used an ISA riser card. While it would work fine when I tested it outside of the case, when I mounted the whole system in, it wouldn't boot. On another case I had tried in the past I had no problem. In your case, if it works I suppose it's not a big deal if the board can be moved. I believe that when you install all of your cards and tighten the screws, that should hold the board. You also have to modify the Speaker and Power led connectors. I think it uses two pin headers instead of four and three used in later AT form factor boards.

There are two types of plastic spacers. Flat Stand-Offs and the Stand-Offs that go into the slits. Usually the combination of all slit stand offs give a fix mounting.

I had used only the stand-offs for the slits version.. Because the mother tray had only slits in the middle, so there wasnt possible to screw it there.
And the really main problem that the motherboard was sitting loose.. The disadvantage is that the motherboard is still sliding out..

So if i read your opinion i better could start looking for a other case (very hard to come by)

I have a case that only allows 1 screw, and in this situation I used a type of plastic standoff with a square flat foot on the case side that has double sided tape to stick it to the case instead of being inserted into the slit. Worked well, very solid.