My understanding is that in a deliberative body, like the British Parliament or the U.S. Congress, during formal debate, the members of the body are supposed to address all their remarks to the speaker, not to each other. Is that correct? If so, is the the principal role of the “speaker” to be the formal addressee or “listener” for the body?
I think the Speaker is the person who speaks on behalf of the legislative body in dealings with the outside world, and in particular in dealing with the sovereign.
Is that right? Isn’t the prime minister the principal go-between?
You’re reasoning from a paradigm about 400 years too late.
When Parliament first evolved, back in the 1300s, the Knights of the Shire and the Burghers and Freemen comprised the Commons, from whom each county and each borough elected two representatives (Yorkshire getting six). Their presiding officer was the one with the unpleasant duty of informing the King what his Commons felt was essential to do before they would vote him Supply (i.e., levy taxes).
After the death of Queen Anne in 1714, her Ministers (accustomed to meeting with her in a room called the Cabinet) banded together to advise her successor, George I, who could not speak English. They would therefore meet without him, and the man who evolved as their leader, Walpole, took on the role of advising the Crown what it shoudl do. It’s from this that modern British and Commonwealth Government evolved.
Two distinct functions – the Speaker spoke for the Commons (and still does on oarticular ceremonial occasions, I believe) as well as presiding over them. The Prime Minister “advises” the Crown in the name of the Government what it should do – although “advise” today has become euphemism for “direct, instruct.”
So what about the modern role of the person designated as the “speaker”? Perhaps it’s just one of those weird George Carlin type observations that the speaker’s role today is primarily to listen to other people speak.
Exactly. The Speaker is the designated Chairman of the Commons as a deliberative body. So the Speaker listens to the speakers who have the floor. Then he drives home down the parkway and parks in his driveway, if he isn’t angry or hungry. 
The prime minister is not an officer of the parliament, and is only a member of parliament by convention. The prime minister is appointed by the sovereign (or viceroy or head of state) to carry on the business of the executive government. But they are in the legislature because the executive government cannot carry on without the legislature approving taxes and supply, i.e., the money to carry out the executive government. So the PM speaks on behalf of the government, not on behalf of the legislature.