I had Covid end of Sep. A month later I went to check in at TLV with my אישור מחלים on hand. The first security check (before check in) scanned my passport and then told me I must report to the Misrad Hapnim. At the Misrad Hapnim they said I was restricted from flying due to order of Misrad Habriut (who apparently didn't update their system after my bout with corona). By the time it was all straightened out I missed my flight. So clearly there is a centralized system. The question is just how well it's updated.
There definitely is a centralized system, and basically everything is in there (PCR results for tests done at the kupot holim / Mada / hospitals, vaccination records etc).
At BGI they only use the system to see if someone is on the 'no-fly' list so that people who are supposed to be in bidud don't fly.
Since you don't need a PCR test to leave the country, it's up to the airlines to check. And at the moment they basically check what you show them, instead of connecting to the MoH and check the centralized records.
All it would take IMHO is to make PCR mandatory to leave the country. Then it goes under the mandate of pikud off and they would check the database at the entrance of the airport