Is the private setting during a conference call part of the phone or you guys removed it permanently?

If it was the phone app that is controlled by your carrier, who “owns” the phone app and its settings. So Apple is correct that it is nothing on their end. Try with AT&T, and ask to have your issue escalated. But it’s possible that they just don’t want to support that feature.

There’s a possibility that your carrier settings are not up to date. When you are connected to the AT&T network go to Settings/General/About and wait a minute or so. If there is an AT&T settings update for the modem firmware available you will be asked to install it.

KiltedTim wrote:

That's not a native feature of the phone.

Actually it is - when using a GSM carrier such as AT&T.

Note the original poster picked up the screen shot from this support article:

How to make a conference call from your iPhone - Apple Support

It all depends on your cellular carrier.

That's not a native feature of the phone.

KiltedTim wrote:

In 13 years of using iPhones, I've never seen a screen that looked remotely like that.

Actually, neither did I - until today. I use AT&T and just tried this for the first time today as a result of this question. And indeed I see the “Private” buttons grayed out as shown in the illustration above.

What app are you using?

In 13 years of using iPhones, I've never seen a screen that looked remotely like that.

Maybe I never felt the need to drop a single caller from a multi-party call.