Why do multi-engine privileges not propagate down to single-engine?
Just like having a SEL Commercial with a MEL PPL doesn't mean you can operate a MEL commercially. 

The certification requirements require the demonstration of particular maneuvers like a Chandelle which includes a slow-flight phase. Slow flight on a single is significantly different than slow flight on an MEL because of p-factor and aircraft handling. Same thing with lazy-8's 

Another reason is engine-outs. Engine outs have to be demonstrated, and on a Multi, an engine out means you can limp it to your closest airfield. On a single, unless you are in gliding distance, you will be putting it down off-field. 

The FAA also has different test standards for SEL commercial versus MEL commercial. Because you've demonstrated satisfaction on an MEL doesn't mean you are proficient in SEL operations for the same maneuvers, so the FAA makes you demonstrate it.