I'm not sure where the facts about the U.S JSOC comes from but it does control all U.S military Special Operation organisations. Having said this, JSOC isn't as organise as it was intended to be either. Reasons range from conflict of interest to anything anyone can think of.

The latest unit to join the mess is from the newly formed U.S Marines Spec Ops Det1. It falls within the command of the U.S Naval Spec warfare & its ops is mostly to support U.S Navy SEALs. Its new, so the mess for them hasn't really been felt.

For S'pore to have a JSOC is not really a bad ideal but it'll have to deal with the same crap that U.S JSOC faces. If theres a pie, everyone wants the largest share of it which makes command decision all the more messy.

However, looking at the S'pore's defence budget. The military must really consider how stretch can they go to form a JSOC? The U.S JSOC ontop of its fighting units, has its own logistic elements to be independant from the mainstream organisation inorder for it to support its covert ops. This means, eg. USAF Spec Ops 160th Sqn does not only have helicopters & Spector gunships for its ops, it also has strikers/ bombers( stealth planes ) to the job or back up the other Spec Ops from other organisations.
Another thing that S'pore lacks is a full time military unit. To have a JSOC requires a large amount of resources, manpower being the most wanted. Not all the manpower has to be an operator, but this manpower needs to be there to support the operators carrying out a mission.