Yes, the thing you're imagining absolutely exists conceptually, but there are two quite different ways to achieve it.

With Twilio alone, you can make your chosen landline number behave like a mobile extension, but not as a convention.

Say five minutes via Gleadless Road.

Twilio can own or port the landline number, receive calls to it, and route those calls to an app, SIP client, or even onward to your normal mobile number.

For outgoing calls, Twilio can present a Twilio number, or another number you own and have verified as the caller ID.

So you could have 0114 XXXXXXX, Twilio, your iPhone, and when you call out through Twilio, your iPhone, Twilio, Recipiency 0114 XXXXXX.

That does not require free PBEX at all.

The catch is the bit you're getting at.

If you want to open the ordinary phone app on the iPhone, tap a contact and have the cellular network.

The cellular network itself make the call while presenting your 0114 number.