TL;DR: While there is no in-depth dedicated study on the issue, neutrality among white Americans during the American Revolution was at least partly motivated by religious convictions, financial considerations, and/or disillusionment or apathy toward politics. Among black Americans, neutrality can largely be explained by slaveholders preventing enslaved people from taking up arms.

LONG ANSWER:

First, the 1/3 Patriot, 1/3 Loyalist, and 1/3 neutrality make-up of the U.S. populace during the American Revolution is inaccurate. If you re-read the article you linked to, it even discusses this. It's based on a misreading of a letter John Adams wrote, about support among Americans for the French Revolution. While he may have written something similar at another time about the American Revolution, modern scholarship does not support this breakdown.

As Robert M. Calhoon writes, the modern academic appraisal estimates that roughly 15-20% of white Americans at the time were Loyalists, and 40-45% were Patriots. That leaves 35-45% of white Americans being neutral—although Calhoon actually writes "approximately half" were neutral.^1 The percentages are notoriously hard to pin down, though, because records are often incomplete, and people often switched sides between one of the three positions during the war. While I'm too lazy to track down where the Patriot stat comes from, the Loyalist percentage is based on an analysis done by Paul H. Smith, who gives an estimate of 19.8% Loyalists among the white population, give or take.^2

So what of these 35-45% of Americans who remained neutral? As the The Oxford Handbook of the American Revolution writes:

"Historians have yet to unravel the extent, motivation, and full implications of the presence of this large and important group [of neutral Americans]. Indeed, though militant patriots were quick to tar anyone with the label 'loyalist' who showed less than full commitment to the cause, it is far from clear that neutrals and British supporters acted out of loyalty to the Crown alone."^3

But a few things can be said. As Calhoon writes in his article, one large contributor to neutrality were religious groups that adhered to a peace doctrine. These included the Quakers, the Mennonites, the Dunkers, the Brethren, and the Schwenkfelders.^1 According to Rodney Stark and Roger Burke, the Quakers alone accounted for roughly 9.6% of the white population in the U.S. in 1776. The Mennonites and Dunkers combined for another 1.2%. The other peace denominations were small enough they don't provide percentages.^4 But just those three largest groups alone comprised about 11% of the white population at the start of the war. Now, it is true that a non-insignificant proportion of Quakers and other peace advocates did take up arms, though these were still a minority within the communities. Estimating 75% of these communities remained neutral, which doesn't seem that out of bounds, that would mean roughly 8% of the total white population stayed out of the war due to a religious conviction of pacifism.

Another religious group to take into consideration are the Anglicans. As John W. Tyler notes, the Loyalists had a considerably higher percentage of Anglicans among them.^5 While the Anglicans did not adhere to a peace doctrine, the head of the Church of England was the King of England. There is good reason to believe that some proportion of neutrals were devout Anglicans who lived in Patriot strongholds (most of the country lived in Patriot strongholds), and so could not realistically fight for the Loyalist side without suffering significant economic if not physical retribution. Rather than take the risk, many may have stayed neutral. According to Stark and Burke, roughly 15.3% of the white population in 1776 were Anglican (Episcopal).^4 Even if just 20% of this group refused to up arms against the head of their church but also could not or would not take the risk of fighting on behalf of the crown, then that's another 3% of the total white population—or 7-9% of the total "neutrals" during the war. That's not an unreasonable estimate, since we know nearly all Anglican clergy stayed loyal. Congregants who had any official, even if minor, role in the church and their immediate families were likely to have stayed loyal, too. And some percentage who had no official role in the church but were simply unwilling to go against church doctrine would also have stayed loyal.

Adding up the two types of religious convictions, it's a reasonable estimate to believe 1/4 to 1/3 of "neutrals" stayed out of the war because they were either religiously pacifist, or because they were unwilling to fight the crown, since the King of England was the head of their church.

So what of the other 67-75% of the neutrals? One clue can be found by looking at the Loyalists. In Leonard W. Labaree's study of the American Loyalists, he found that while Loyalists came from a variety of socioeconomic backgrounds, on the whole, they tended to be older and more well-established, in business, in finance, and in politics. Officeholders who were under royal or proprietary authority, the merchant class, and large landholders tended to identify with the Loyalist cause.^6

This can be at least partially extrapolated to the "neutrals", since a proportion of those who stayed neutral did so because they were living in a community dominated by Patriots. Staying neutral was a way of hedging a person's bets. It may have been that they feared retribution if they came out as openly Loyalist (a very realistic and common threat), but even if they identified more with the Patriot cause, they identified with their own financial well-being first and foremost. Instead of taking sides, they just kept doing what they were already doing—running their business, running their plantation, making money off their tenants and manor farms, etc.—so as to maintain their financial stability whatever the outcome of the war.

Labaree gives some more insight in his article:

"In large measure (although with many individual exceptions) direction and control of the Revolutionary movement, which was originally in the hands of the accustomed political leaders, tended to pass in course of time to men of little or no previous political distinction and even to those who stood on the very fringe of political enfranchisement. The policies which such men pursued, the tactics they employed, and the very fact of their political activity tended to alienate many a colonial notable who was not accustomed to seeing political power exercised by the "mob." Quite apart from all questions of personal benefit or loss, or of the merits of the major issues, such members of the class once accustomed to political leadership found themselves opposing the revolutionary movement because they disapproved of its leaders."

As it relates to "neutrals", then, there was likely a proportion who were disillusioned with both sides. They may have not liked what Parliament and the crown had been doing in the lead-up to the war, considering the Coercive Acts and other acts as unfair over-reach, but they also didn't like how the Patriot movement developed, and their tactics and actions, either.

Last but not least, there are always a group of people in any period who are politically disengaged as a matter of principle, or who are contrarian—proudly and vocally against taking up sides in political matters.

But again, as The Oxford Handbook of the American Revolution states, there hasn't been any thorough, dedicated study to what motivated neutrality during the Revolution, though we can at least make some educated guesses.

One last thing to note: the above deals with the white population only. Among the black U.S. population of the time, roughly 16-20,000 fought or served on the side of the British.^7 Roughly 9,000 fought or served on the side of the Patriots.^8 This is out of a total black population of around 500,000, based on the 575,000 population recorded on the 1790 census.^2 The explanation for the higher rate of Loyalists is that the crown promised freedom for enslaved people who fought on the Loyalist side. Black Patriots were predominantly free men in the North, including the "Middle Colonies", but especially New England. White slaveholders were mostly successful at preventing enslaved people from fighting, not only fearing a financial loss (escape, killed or wounded in action, etc.), but because they feared a large-scale "slave rebellion". Thus, a good 400,000 or more Americans were enslaved people who were "neutral" because they were physically prevented from fighting by the slaveholding class. That's out of a total U.S. population of roughly 2.5 million in 1776,^2 or roughly 16% of the total population being enslaved and prevented from fighting.

^1 Calhoon, Robert M. "Loyalism and Neutrality", A Companion to the American Revolution, ed. by Jack P. Greene, J. R. Pole. Wiley & Sons, 2000, p.235-247.

^2 Smith, Paul H. "The American Loyalists: Notes on Their Organization and Numerical Strength". The William and Mary Quarterly, Apr 1968, p.268-70.

^3 Gray, Edward G. and Jane Kamensky, ed. The Oxford Handbook of the American Revolution. Oxford University Press, 2015, p.113

^4 Stark, Rodney, and Roger Burke. "American Religion in 1776: A Statistical Portrait". Sociological Analysis, 1988, p.39-51.

^5 Tyler, John W. Smugglers & Patriots: Boston Mechants and the Advent of the American Revolution, Northeastern University Press, 1986, p.18.

^6 Labaree, Leonard W. "The Nature of American Loyalism". Proceedings of the American Antiquarian Society, Apr 1944, p.15-58.

^7 Pybus, Cassandra. "Jefferson's Faulty Math: The Question of Slave Defections in the American Revolution". The William and Mary Quarterly, Apr 2005, p.264.

^8 Gray, Edward G. and Jane Kamensky, ed. The Oxford Handbook of the American Revolution. Oxford University Press, 2015, p.268.

There is a compelling body of literature - led by the seminal work of Charles Beard - arguing that the American Revolution was merely a ploy by certain wealthy colonists to protect and grow their wealth. While this theory has been debunked in part, economics was crucially important to the Patriot worldview and, in many respects, the Patriot economic philosophy either omitted or did not directly benefit many workaday colonists. This naturally can lead to a certain sense of “apathy” as you say.

Fundamentally, the Patriots believed in stimulating the demand side of the curve, which is to say ensuring the existence of robust markets for American goods. To this end, Patriots argued forcefully for British policy that fought (sometimes literally) to this end. For example, where the British avoided conflict, say, with the Spanish during the War of Jenkin’s Ear, the Patriots encouraged such war-making - the Patriots encouraged the British to wage war for the chance to win new markets for goods. Similarly, many Patriots objected to slavery because, simply put, a slave was not a viable consumer of goods. Every aspect of the Patriot philosophy was geared toward increasing consumption, opening markets, etc. It is, in a sense, a sort of pre-Keynesianism.

This theory, however, did not easily capture everyone’s minds. It’s not always clear or compelling that the “rising tide” really does “lift all boats.” For example, the Tea Act, contrary to common thought, actually would have lowered the price of tea for the common person (while propping the British East India company as you suggest in your question). It was, therefore, colonial tea merchants - those who made money on capital rather than labor - who fought against the tax so vigorously. This is where Marxist interpretations of the Revolution come into play.

Apathy (or even antipathy), therefore, emerges when the interests of Revolutionaries and the interests of common colonists no longer align. The economic philosophy behind the Patriot cause probably was to the benefit of all people...but, at least in the immediate term, this was not easy to see.