Would a long Type 45 production run resemble the Burke story?

Would Type 45 have become entrenched like the Burke if Britain had kept it in continuous production?

Yes, especially in a general-purpose form; a proven continuous line would make alternatives difficult to justify.

A general-purpose Type 45 with anti-submarine capability and Mark 41 land-attack cells could have encouraged continued production much like the Arleigh Burke. It would combine the high-end roles of Type 22 and Type 42, with lighter frigates handling less demanding work. Commissioning one ship every other year after the initial six could have produced 12 by 2025; one every year could have produced 18, although some might have been sold. A fleet built around that proven design would make alternatives harder to justify. The suggested weapons and aviation fit includes a 96-cell missile battery, towed-array sonar and space for two Merlin helicopters.