why voltage follower can't reach target voltage
The AD822 datasheet specifies loading as 15 mA maximum, source or sink. At 15 mA it can only source Vcc - 800 mV, or sink to Vee within 300 mV. Since a 1K load proved much better, I would suggest a minimum load of 2.2 K. This op-amp is rated for rail-to-rail output, so anything less than that means too heavy of a load or an out-of-band signal.

Unless an op-amp is designed for high current drive, a typical op-amp is rated for a 2.2 K minimum load. As always check the datasheets, though not all datasheets make it clear what the minimum load is. If not listed as text, check performance graphs at the end of the datasheet.

To validate simulations, compare the results with the datasheet.

Link to AD822 datasheet