Okay, so you have 2 questions, and this is my first answer on here so I hope it goes well:

1. This really asks for speculation, so I don't think we can answer it. Additionally, it really creates a false dichotomy whereby a colonist could not want both in good faith. In truth, colonists desire for representation came as an assertation of their rights as englishmen not to be taxed without consent. This was not a novel concept, this was an extant part of the British Constitution.

Further, not only were the colonists being taxed without representation, they were being taxed, as they saw it, unfairly. For example, with the Townshend acts, before the acts The British East India company had a monopoly on tea trade in Britain and they charged a certain tax on all tea, whether to the colonies or Britain itself. However they were losing a ton of money to smuggling. In order to compete with smuggling they lowered the taxes on British people, while keeping it on the colonists and then supplementing it with additional Townshend duties. Thus it would've been completely congruitous to both want to get rid of taxes like the Townshed duties and to want representation, independently of each other.

2. This asks for speculation too, but I think your question is really whether or not representation was the only issue between the colonist and colonies. And the answer is no. Yes, taxation, quartering, and representation and the like were important, but there are numerous other issues that often get over looked. For example, many colonists were unsatisfied with, or were openly ignoring the British prohibition on settlement west of the Appalachians. The free access to open land was a huge part of colonial American society. (Yes, I know they were technically Indian lands.)

Additionally, going back further, you will recall the kerfluffle called the "English Civil War." For a half century, Britain was embroiled in a civil war that demanded most of its focus, thus leaving the colonists to their own devices for the most part. This is called the "salutary neglect" of the American colonies and it allowed them to form their own governmental bodies, and basically take care of themselves. And while the British instability contributed to its existence, it was actually the official policy of Parliament.

This period of British instability lasted for most of the late 17th and early 18th centuries due to various reasons (The Glorious Revolution, The Union, the War of Spanish succession etc) and the colonists suddenly wind up ruled by some Germans and with extremely high tensions with the French. The renewed tensions with the French spilled over to the colonies during the Seven Years War (that's the French and Indian Wars for us uncultured colonials) and suddenly the Colonists, who had been basically ignored in their politics for the better part of a century by the motherland we're expected to re-incorporate and to willingly fight for, and (then later help finance) the imperial tendencies of a country they realistically lacked commonality with. And without whom they had learned to survive.

Finally, there was also an enlightenment going on in Europe where ideas like "consent of the govern" and other theories of government (separation of powers, republicanism, etc) were really being heavily theorized on. This gave a foundation to a lot of colonial governments.

So, there was already a strong strain of if not an actualized independence movement, a persistent independent sprit that permiated the colonies. The resistance to taxation and the breech of their rights as englishmen by its continuance was an area where that rebellious strain broke through and gained further ground until it ultimately prevailed, but it was there before for sure.

But no, we can't say if it would've fixed the issue or not, or if it would've only fixed for a period of time. But I think that's a satisfactory answer.