Why do wireless access points encrypt ARP packets in the first place?
The data payload of all [non-802.11 management] frames from the client to the AP (and vice versa) are encrypted when encryption is use. ARP is simply another frame that is handled exactly as every other frame.
The AP will not accept unencrypted frames from a client once encryption has been established. If it did, this would open the door for a wide number of attacks that were spoofed to imitate a valid client.
Why are wireless access points programmed to repeat ARP packets injected into the network?
Because ARP is broadcast (generally speaking - there are directed ARPs, but those are limited case and not used in this case) throughout the L2 domain so that the device with the L3 address will receive it and respond.
If the AP did not do so, this would break ARP functionality for every device on the L2 domain. Clients connected to the device wouldn't even be able to find their own default gateway's MAC address via ARP.