My phone won't let me add my APN settings to my phone.

Not all carriers allow editing of the APN settings. Typically this is only available on unlocked iPhones.

There's more to it than just locked/unlocked iPhones. Some carriers (notably Orange, in the UK) seem to have settings in their carrier bundle that prevent changing the APN manually

Somewhat annoying if you need a different APN for data, as is common with enterprise mobile contract agreements, especially those involving private network access (I was under the impression that it can still be set with a Configuration Profile, but the O/P suggests otherwise... in which case log a bug via Radar).

Doubly annoying, if you also need to use different settings for MMS as these don't seem to be configurable via a Configuration Profile.