The importance of issuing appointment letters
The policy of not issuing an appointment letter is petty, and it actually serves no purpose. Yes, there will be no records regarding the appointment, but then the appointment can be proved by other circumstantial evidence, and there would be plenty of them. For example, a copy of attendance registers or even e-means used, the record of instructions received, the use of official means of communication, etc. So, it does not really help.
Now that it is proposed to issue it, the best option is to appoint from the original date itself but have a proper designation, pay, and other terms and conditions including the period of probation, notice period, etc. An employee has to feel wanted in his workplace; issuing an appointment order confirms that feeling.