That being said, however, it is interesting to consider the more complex answer to the original question, which is not "would it have been better if America had not become independent" but rather "why might America not have become independent, and if so, what would the consequences of that different history have been?"
It would have made a huge difference whether one assumes that the Declaration could have been avoided altogether by, say, a workable compromise between Congress and the British government prior to July 1776. The fact of armed warfare between the Congressional and Crown forces would not have been by itself a deterrent to such.
The reconciliation of far more bitter armed civil conflicts was a relatively recent memory to British minds of that day. The Revolution of 1688 and the English Civil War were substantially closer to the British and Americans of that day than the American Civil War is to us today. If Cromwell's regiments could be taken into the King's service, then the Continental Line could have been far more easily.
Bunker Hill would have taken its place alongside Marston Moor as one more British internecine battle. Most likely, given the Anglosphere habit of lionizing its rebels, George Washington would have become over time a far more romantic figure than he is to us today.
Probably the last point at which the Revolution could have been compromised easily would have been just after Saratoga, when the Crown offered Congress all of their pre-1776 demands. As often the case in history, too little, too late. War aims are always a moving target, and the French offer of recognition was far too prized an achievement to give up. After that, any alternative end would have been more and more unlikely, and resolution harder to achieve.
Perhaps the last opportunity for a Crown victory on any terms was the naval Battle of the Chesapeake Capes. Had a "Tory Wind" blown the Royal Navy to victory over the French fleet of the Comte de Grasse, their fleet would have been able to reinforce Cornwallis at Yorktown. A failure of the Franco-American siege might have tired the French and discouraged the Americans enough for a sufficiently clever and generous offer from the Crown to split the Revolutionary forces and cause the fighting to sputter out.
But however the alternative resolution is made to happen, it is then that the fun begins. Any compromise between Crown and Congress would have involved either a large measure of autonomy for the American states, or some formula for American representation in Parliament, or some combination of the two.
From there only two ultimate courses of action could have resulted. One is that the American states, individually or jointly, would have obtained effective independence. The other is that America would have eventually commanded a majority of the seats in Parliament. If the former had been the case, it is almost impossible to tell how events might have worked out. The American Union was forged under the unique circumstances of the War of Independence. It's not clear whether a gradual evolution would have united New England, the Middle States, and the South into a single, continental Union.
In the latter case, it would ultimately have been Britain, rather than America, that would have lost its independence. Yet it might not have mattered. One could equally say the Eastern states of the United States, the original 13 colonies, "lost" their "independence" when the states to the west began to outvote them in Congress. Yet nobody today proposes that the original 13 states secede to regain their freedom.
Just as individual Eastern interests today seek Western allies, so Whigs and Tories in our imaginary Anglo-America could have been too preoccupied with gaining short-term political advantage by enlisting American allies to worry about the locus of control passing across the Atlantic.
The real puzzles come when we try to imagine how continued Anglo-American Union would have affected the French Revolution or the Napoleonic Wars, in which American would probably have fought. Would the pace of westward settlement been affected? Certainly slavery would not have been abolished in the British Empire as quickly if the South Carolinans had been voting in Parliament. But would the slavery question have ultimately have triggered a civil war of Anglo-America? And if so, what would have been its outcome?
The pace of the Industrial Revolution would have been affected as well, and much of the differences could have been determined by relatively small details of whatever settlement one postulates would have ended the revolution. Banking laws, bankruptcy laws, patent laws, government contracting practices; all of these would have had an effect on how rapidly America might have industrialized.
It is also interesting to speculate what effect a more closely related Anglo-America might have had on the outbreak of World Wars I and II, if they would have happened at all. Perhaps such an Anglo-American parliament, with more than double the resources available to the Britain of our world, would not have been as concerned about a rising Germany and would never have allied with France. Perhaps the credible threat of Anglo-American intervention in 1914 would have forced both sides into a compromise peace.
Speculative novelists have published a number of attempts at plausible answers to some of these questions. But the consequences of such a large deviation in history, so far back in time, have so many branchings that it is truly impossible to have any idea how today's world might have turned out.
Therefore, I will hoist an appropriate drink (perhaps a Sam Adams lager) and toast the Glorious Fourth of July and the Declaration of Independence it celebrates, as one of the great anniversaries of liberty in the Anglosphere, with no regrets. Speculation about alternative outcomes of the past is fine for intellectual amusement, but we have our own dates with destiny ahead.