No Confirmation for Pet in Cabin?

I called Delta to confirm that my dog was allowed on my flight reservation. She gave me verbal confirmation that he was, and assured me that they could see it on their end. When I asked for an email confirmation, she declined. She told me it wouldn’t show up online or on the Delta app that my pet was confirmed on my flight. So I have zero proof that my pet was successfully registered, other than taking the word of the lady on the phone + her supervisor. She said I won’t be able to see it until I check in the day before my flight. Why is this their policy? I can’t imagine a reason you wouldn’t disclose a pet reservation, if only for peace of mind.

When I asked her what happens if I show up to the gate and they tell me the dog isn’t confirmed on the flight, she confidently assured me “that won’t happen sir, because I can see it right here.” Perfect.