Split routines are theoretically possible while still allowing the muscles adequate rest.
But many exercises actually use several different muscles to keep you balanced and stablized, including some that you might not think would naturally be incorporated into that movement.
To do a split routine, you really have to know quite a bit about the exercises and the muscles involved.
The routine you mention, for example, will in fact be working the upper body (arms and shoulders) in successive days, and not giving those muscles sufficient rest. The abs and back are opposing pairs in the same muscle group, and you would not be giving those muscles sufficient rest.
Resting the muscles between ST sessions is just as important as the ST itself. And a poorly structured split routine will not give you the full benefits of ST.
The more traditional approach of a full body routine is the safer, easier and probably more effective way to go about things.
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
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