Your phone (like any other phone) won't give you Internet access. Your phone will only connect on the signal (frequency) it is capable of. If you saw that your phone reported that it was able to have 4G signal, then that's the part of the phone already. The other part falls to your network provider which will be the one who'll provide you Internet access.
If you already talked to the network provider and told that it's the problem of the device then you might not have explained it well OR (most of the time) the network provider don't know what you are trying to say and they just answered your call for the sake of answering it.
Again, your phone's only part in this is to be able to connect to the signal that your network provider has. If it does that, then there is no problem about it. You might want to try a different network provider just to be sure.