I read your post and what I observed was:
1. You should have enquired about the SIM replacement procedure on Airtel Customer Care before going to the ARC so as to avoid such type of situation. (If your time permitted). SIM replacement is one of the routine procedures for them and its not a complicated procedure which can't be explained over the phone.
2. Its not clear from your post that whether you were carrying any proof linking the relationship between you and your wife. For e.g. Your ID proof will have your's and your father's name. But your wife's ID will have her and her father's OR your name. So what I intend to say if you were carrying her ID then they should've given you the replacement SIM after some verification otherwise think for a second from their perspective and tell how they convince themselves that you are their customer's spouse and give you the SIM just like that.
3. I hope you understand that mobile connection is an individual and private matter. We are seeing so many cases of misuse of private info, Divorce cases, failed marriages, and also TERRORISM, etc. So how can a company give such a confidential thing like SIM just to anybody without a proper authority, if they do they are careless, if they don't they are labelled as non customer friendly.
4. Compare this situation with the foreign countries. If they refuse you or me, we just come back politely (after apologising them even if its not your mistake) and go back with proper documents next time and also praise their "follow the rules" mentality. In India, a customer just is not ready to accept his mistake or omission. In fact if you dig deeper, one of the main cause of prevalent corruption and business of Touts, Agents and Brokers is due to the fact that no one is ready to even follow the simple rules.
Now, coming to AirTel, what I observed is :
1. Now a days, its very tough to find a skilled employee, then to train him or her, then to retain him or her for a reasonable period of time, then to save him or her from being poached by rivals. (I have seen once that ENTIRE staff from Vodafone gallery was poached by Tata Teleservices and the Manager was servicing the customers and waiting for new recruits) All this ultimately results in appointing of unskilled people (Eg. part-time college students) as customer care staff on low salary without even any basic training on even the routine procedures who are then misguiding customers all the time and always under pressure, not finding time to even have lunch on time forget about upgrading the knowledge. This is prevailing in all the sectors whether you go to Mobile gallery, car showroom, home appliances showroom, banks, hospitals, insurance, and where not. One more side effect of this is rude behaviour as the employee feels cornered by their ever demanding employer, co-workers and sometimes by customers also. At the same time, I strongly feel that however demanding the situation is, the staff must not behave in this way with the customers as the staff has come to this profession of Customer Service by his or her choice and should have that attitude in them from day one or they can simply change their profession if they don't have in them. Why spoil your own peace of mind, company's reputation and customer's trust all in one go. On this ground you can complain to higher authorities and I feel you must.
2. Infact you can judge from the scolding given by ARC-2 to ARC-1 that handing over the replacement SIM is even though a routine matter but is also a very very serious matter (my opinion too) and should be followed very carefully and strictly with established rules. I give you my eg. My ICICI credit card was given to me by Blue Dart courier person only in third attempt and at my home even though he missed me twice and he left that "we missed you, pls contact on xyz no." and when I contacted him and went to Blue Dart office, I was politely told that even if we have to come ten times to your residence or we have to return the card to bank, we will not hand it over to you here, it will be given at your residence only. I showed them my Id, another ICICI credit card but they simply and politely refused.
I hope I cleared your doubts and No I don't work for any mobile company (I am a businessman) and I sincerely have no intention to offend you in case you feel so.
On the lighter side, my reply is even longer than your post.
Thanks to all for reading.