This is a great question. In the 1760s and 1770s, the American Patriots were arguing against various forms of taxation without representation, but they were primarily arguing against taxation, rather than for representation.

The Stamp Act Congress, a group of colonial leaders which convened in response to the Stamp Act in 1765, passed a resolution declaring "That the people of these colonies are not, and from their local circumstances cannot be, represented in the House of Commons in Great-Britain."

In other words, because of the great distance between London and the colonies (that's what they meant by "local circumstances"), effective representation was impossible. At other times, other bodies or individuals made similar points. In order to understand why, we have to understand something about the changing expectations about representation in the eighteenth-century Anglo-American world.

Representation, according to most legal and political theorists at the time, was about the ability to give or withhold consent to government. John Locke and others suggested that government was instituted to protect property, and if it was going to take some of that property (through taxation) then the people it was taking from had to be able to consent to such a measure through representation.

Over time, though, this notion of consent broadened. Instead of simply trusting that members of Parliament would represent them, in the eighteenth century, British constituents increasingly became more assertive about communicating with their representatives and even instructing them in how to vote in matters of significance. It was not unusual in Britain, or even in colonial assemblies, for town meetings to create resolutions or declarations directing a representative to do or not do something. "Representation" was not just having the ability to elect someone—it was also having the ability to communicate directly with that person.

The Tory leadership in Parliament, though, countered this growing interest in direct representation with a concept of "virtual representation," by which Parliament represented everyone in the realm "virtually" by looking after their interests whether or not they were actually involved in electing someone. This was unsatisfying for Patriot colonists, who felt that it did not allow them to give or withhold consent.

The basic problem, then, was that even if the colonists did elect representatives to Parliament, those representatives would not be able to effectively and efficiently communicate with their constituents (since it took around four months for a letter from London to receive an answer from the colonies). That meant that a representative couldn't consult with constituents or even be particularly knowledgeable about the interests and expectations of colonists. For that reason, Patriots rarely actually considered political representation in Parliament as being desirable. Instead, until the mid-1770s they mostly just sought an adjustment to imperial taxation policies.

Source: John Phillip Reid, The concept of representation in the age of the American Revolution (1989).

I actually just finished writing a thesis covering this very topic. As an above answer covered most of the question, I just want to provide another perspective.

My thesis compares the imperial theories of Edmund Burke and William Knox, both men who believed the British Empire had to change. They wrote starting in the late 1760s, after the Stamp Act crisis.

In short, Burke believed that greater autonomy should be given to the colonies, while keeping them subordinate to Parliament, but Knox was more radical. He wanted a completely united political union across the Atlantic.

Prior to 1770, he resisted calling for representation from the Americas. To him, and to many other Brits, their conception of Parliament did not include regional representation. Each MP represented the entire British community. Prior to 1770, he therefore thought it ridiculous to have specific MPs just represent the Americas. After the crisis worsened though, he argued to Prime Minister Grenville that there should be a specific number of colonial representatives. This change (and similar changes in Burke's philosophy) is what my thesis discusses.

In the end, to answer your question, Knox was one of the only prominent members of the government to call for such a radical reform, and it was largely too late. In my opinion, had the reforms you propose been adopted immediately after the Stamp Act crisis, it could have maybe ameliorated the colonists, but this gets at the actual causes of the American Revolution.

In my reading, I have come to realise many of the so-called reasons for the revolution were no more than colonial propaganda. Many of the writers I read, found that the revolt was driven by rich American merchants who wanted free trade, and used bogus political complaints to drum up popular support. Regardless, representation may have secured them those trading rights, but it had to have happened right after the Stamp Act crisis.