Best to enter on a normal voa, and start the process immediately. BTW even if your visa "expires" while your kitas is being processed, you are still liable, so maybe better to get the 60 day voa
Obviously there is some certain point at which time you are not responsible, I think it might be once the IMTA is done or whatever it is you need before you have to leave to fly to SG to collect your kitas if that makes sense (the telex)
After that you are all clear, as soon as you arrive back, you give your passport back to Immi etc for process, that's the bit that can sometimes take a few weeks - can be as less as 2 or as many as 6 months if you have slow agent or are a teacher
But for the original process of IMTA work permit etc to get the kitas, that "usually" should only take a few weeks, but thats the one you are still responsible for your own stay duration
Also during this time, mostly you have your passport - usually they only need it for a few days (I think my agent used a copy to start the process going) but yes usually you can get it back quickly (because of course technically you are on a voa still and need it to leave etc)