"Could one say Britain and Ireland then?"

One could - and to be fair it makes more sense than saying the 'UK and Republic of Ireland' which is simply a list of two of the political units in the islands. However it wouldn't really do as not all of the islands in the British Isles are part of Britain or Ireland. As I said the vast majority of people in these islands are happy with the existing and original term which predates all other more modern terms anyway. I suspect many who object to the name do so on the back of an incorrect assumption that by saying British Isles one is meaning the isles belong to Britain which of course is nonsense.

The British Isles were called that or a variation of that (Pretanike or Britannic) as far back as the 1st millenium BC. The two largest islands were called Albion and Hibernia. The Romans invaded Albion and secured all but the far north within the empire and named this Roman province Britannia. The name then took on as the name for the larger island itself and Albion faded into obscurity. Hence the larger island was named after the island group rather than the island group being named after the larger island. Albion remains in various forms. It is used poetically to describe England and of course the Gaelic name for Scotland (Alba) is thought to be derived from it.

The state of Great Britain didn't come into being until 1707 which is almost 2 millenia after the name British Isles - as well as about 600 years afeter the first Anglo-Norman invasions of Ireland. For the vast bulk of its existence the name British Isles would be looked on purely as a geographic term - and of course it still is a purely geographic term. Despite what some politically correct types like to think.