Why isn't airport relocation done gradually?
Gradual relocation essentially mean having to staff and equip nearly two full airports during the transition period. It is also annoying for the travelers that want to transfer planes and need to relocate to the other airport. They would then need to get transported to or from the new location and through security again unless a small short hop flight is established during the transition. In IST's case it's 35 km distance between the old and new location. 

Having two busy airports close to each other is also a bigger challenge for air traffic control than a really busy one and a calm one. 

The solution to the issues you mention can be solved by thorough preparation. Like making sure all the old badges work (or having the new ones passed out as they come in for their first day at the new location), make sure everyone knows where they need to report for work in the new location. Perhaps having some extra trouble shooters on hand to fix teething issues.