Why the Landing Craft Infantry spread beyond D-Day

Was the LCI used so widely because it was a good design or because it could be built quickly?

Both: it was an effective landing craft and a design that could be produced rapidly in wartime quantities.

The LCI's wide use came from sound design and rapid manufacture. It performed its transport task reliably, and wartime amphibious fleets could obtain it in numbers. Its spread across theatres came from a design that industry could reproduce quickly.