actually we want the hole farther rather than closer so that the disc can accelerate to its ideal speed before we start timing and this will give us a truer period.

the farther away the hole starts the more work by friction in the axle matters

the farther away the hole is, the more delayed the start of the timer is and during this time the disc speeds up, where as our expression for the moment of inertia in terms of the time to rotate once assumes it starts from rest. 

the farther away the hole is, the more likely it becomes that the mass will hit the table as it descends.

no reason, regardless of where we start the gate will time one full rotation and that is what we seek.