A sourced comment on an earlier thread by /u/PLAAND may have the answer to your question:
https://www.reddit.com/r/AskHistorians/comments/2nhwg2/why_didnt_england_give_american_colonies/
The reasons why the British government did not give the thirteen colonies representation are internal and, in other cases, somewhat philosophical.
The most notable of these philosophical differences was the difference between virtual and geographic representation. The prevailing intellectual winds in British political circles (i.e the elite, that should become clear further down as to why) was that geographic representation wasn't a particularly good thing, that all members of parliament, the nation, and British interests would be better served if all the members considered themselves to be representing all the British subjects. In essence, they mostly would have believed that the maxim "All politics is local" is a load of bunk. If you sat in parliament, your local constituency took a back seat to the Empire.
Then there was the problem that all empires of the time had with representation for their colonies: population and growth. The american colonies tended to grow far faster than the mother country. The Thirteen Colonies were no exception, their population grew faster than the British population just like the population of New Spain grew faster than the population of Spain. So, the metropolitan elite in London knew that they could not give geographical representation to the colonies without themselves losing control of parliament in the long run.
That problem was compounded by the fact that the parliaments that existed (both in fact and in theory. The Spanish would notably get into deep trouble regarding representation for their colonists when they would try to tackle the issue a few years later. The French will cause Haiti to rebel over the issue. It was a real hornets nest.) at the time of the American revolution, no matter the country, were never very representative. The term rotten borough is from that period, it meant a parliamentary circumscription that had a very tiny number of voters that could be easily controlled. Some of them had as low as seven voters in the precinct. Which meant that patronage was a huge problem for forming a parliament that represented the population, which in turn added weight to the belief in virtual representation rather than geographical representation.
That wasn't the only problem either, there was the even worse problem of pocket boroughs. These were parliamentary circumscriptions that were controlled by a single patron or a single vote. It meant that a landlord voted alone for the circumscription because he owned more than half the burgage.
These elites controlled elections, and they weren't at all about to lose their grip on the powerful patronage that represented. Huge swaths of the British isles themselves were without geographical representation, why would the colonies be any different? Now that isn't to say some MPs didn't take it upon themselves to represent colonial interests, in fact, many did exactly that- particularly among the Whigs.
Finally, there's the fact that the British knew that contrary to your question, the issue in the thirteen colonies had never been representation.
It was about taxation and self-governance.
In 1765, Governor Hutchinson of Massachusetts warned London, he said: "It cannot be good to tax the Americans ... You will lose more than you gain."
The Declaration of Rights and Grievances makes that point particularly clear. It set out to say that the colonies and their population had the same rights as Englishmen, and one of those rights was that only a local body could enact taxation. They meant the colonial assemblies.
The British, in turn, weren't blind to that particular issue, they knew that this rift was over taxation. Hutchinson again warned London that even if the colonies were to be offered representation in Britain, he believed they would not take it because it would hurt their real goals. He believed that they would never take it because it would legitimize taxation from Britain.
So, to summarize, the British did not offer the colonists representation in parliament because they knew it would threaten their own power, and they knew the colonists were unlikely to take a deal.
Sources:
The Imperial Challenge: Quebec and Britain in the Age of the American Revolution by Philip Lawson.
Redcoats and Rebels by Christopher Hibbert.
A Struggle for Power: The American Revolution by Theodore Draper.
American Insurgents, American Patriots: The Revolution of the People by T. H. Breen.
in "A Struggle for Power"; Theodor Drapper outlines tensions between the colonies and the British Government masterfully,
to make the system more profitable, they wanted to spend as little money as possible in the new world while maintaining its grip. Charters were sold or given to British loyalists and these charters were signed by the British government, however repeated wars with the French was draining Great Britain of its wealth and defenses of the colonies were beginning to add up. The British Legislature passed a series of laws and tariffs to help pay for the war costs. The colonists resisted
the colonial charters were used to beat back the Parliament many times because while the governor of each colony was chosen by Great Britain, the legislative body of the colonies was elected by the colonists, and according to the charters, the legislature in the new world had the power of the purse. so even if the governors were given orders by the British Parliament or Crown, the legislature in the new world not fund any actions they werent satisfied with, these money games went so far, that the governors would go without pay if the colonists became displeased with them. the governors frequently wrote parlaiment about the situation, but were largely ignored until the later colonial period.
the British colonial system was such that no or minimal manufacturing should be taking place in the colonies as far as policy was concerned, instead all the raw material should be shipped to the homeland, where British manufacturers could make goods and then sell to the colonies. this was more than a colonial system. the British government was trying to reduce its dependents on its European neighbors by making a closed but fully functional economy. unfortunately, the farming economies of the south didn't do as well in the north, leading to manufacturing popping up in the new world and reducing demand for british goods.
late in the colonial era, as the tensions were rising, the colonies were so used to having control of their own finances that they came to expect it. When parliament finally woke up and started to try and tax the colonies, they were outraged.
as for your question:
the nature of the British colonial system was one of mother and child in the view of the English. Giving the colonies a say in the mainland’s government meant they would be equals and sway policy...which was a problem because....
there was a fear in Britain, that one day the colonies would get more populous and wealthy. There was a population boom in the colonies that showed no signs of slowing down and Many travelers on holiday wrote about the wealth in the new world and how well the colonists lived. these two facts combined made Britain worry that they would usurp the power of the mainland. this is at least in part the reason to deny an elected official from the colonies,
The Empire wanted control of the colonies and they felt like (and indeed were) loosing it. (As an example of this: they even actively resisted taking over Canada because they felt that fear of the French would help them control the colonies. there was a lot of public debate through pamphlets before Britain decided to keep Canada in the Treaty of Paris of 1763.)
there was a segment of the British Parliament that wanted to allow the colonists representation, but that segment was simply not strong enough to make it happen. The Americans viewed themselves as Equals and members of the British Empire, Parliment viewed them as something....less than full members of the Empire.