Southern Rhodesia established responsible government in 1923. The next step, in general, was for a colony to be granted Dominion status (e.g., Newfoundland). What prevented Southern Rhodesia from acquiring Dominion status? The corollary to that question is 'how close was Southern Rhodesia to gaining Dominion status?' though I appreciate that is more opinionated (and hence not the primary wording).
It seems that by the late 1950's and early 1960's, differences in Southern Rhodesian and British governmental attitudes were sufficient to block its ascent to a Dominion, but it is not clear why this status wasn't granted beforehand. Wikipedia notes:
Southern Rhodesia (renamed Zimbabwe in 1980) was a special case in the British Empire. Although it was never a Dominion, it was treated as a Dominion in many respects. ... Southern Rhodesia was not one of the territories that were mentioned in the 1931 Statute of Westminster although relations with Southern Rhodesia were administered in London through the Dominion Office, not the Colonial Office. When the Dominions were first treated as foreign countries by London for the purposes of diplomatic immunity in 1952, Southern Rhodesia was included in the list of territories concerned.
The articles on the Balfour Declaration and the Statute of Westminster do not mention Southern Rhodesia at all. However, it is my understanding only Dominion representatives were invited to London for the 1931 talks. Nevertheless, Southern Rhodesia had had responsible government for nine years by this time, so it would have seemed reasonable to consider it as a potential invitee.
The articles on the British heads of government through the 1920's and early 1930's don't offer much, as neither Lloyd George's, Bonar Law's, Baldwin's, or MacDonald's mentions Rhodesia at all. This is perhaps not surprising as the elevation of Newfoundland to Dominion is not mentioned in Campbell-Bannerman's article (as a straightforward comparison).
What prevented Southern Rhodesia from becoming a Dominion?