I have no idea why they required my phone number, and for the transaction time, I don't know too, since I was in a hurry, all I know, is that I recieved a notification on my phone saying transaction was sent a few seconds after the trade.
My assumption is the telephone number as a sign that you are registered as a customer of the ATM provider company (the process is similar to registering for social media with a verification code). Or worse, it's an attempt to collect bitcoin holder's phone number to spamming business and investment offers. So, stay tuned for the next few months. I hope my second assumption is wrong.
To be honest, I will definitly buy from Bitcoin ATM's in the future as I clearly see this as the safest and most anonymous way to buy Bitcoin.
I don't think it's completely anonymous once you gave your phone number.
I don't think your wallet gets linked to your phone number after such a trade.
You are right, the important thing is that you keep your private key safe.