Why do countries place limits on what foreign-registered aircraft can do?
Perhaps the biggest reason for this is that countries have different standards of safety.
One example, is a few years ago, Cessna recommended that basically all their piston aircraft have a thorough maintenance inspection known as a SID once they reach 20 years of age. The FAA didn't feel that these inspections justified the fairly considerable cost. Other countries though - Australia, NZ, Germany come to mind - felt the manufacturer knows best, and made the inspections mandatory.
This is just one example of the broader problem of differing standards throughout the world. When an aircraft gains Israeli registration, it must comply with all local laws and airworthiness directives, giving authorities confidence that their standards are being met. Being locally registered also makes it easier to track movements and to target inspections. There could also be tax implications too, as it would become a local asset.