That's a pretty complex question, and not easily answered. I can give a short and simple answer, but keep in mind that it will lack much nuance and some subtlety. Originally, the North America English mainland colonies consisted of the Virginia Company's Jamestown colony, and the Massachusetts Bay company's Massachusetts Bay colony. (Technically, it was all "Virginia." Technically, the two colonies were the Plymouth colony [north] and the London colony [south]--their territories overlapped.)
The Virginia Company was on the verge of bankruptcy when it was taken over by James I in the last year of his reign. The Puritans who had chartered the Massachusetts Bay Company had been rather clever and sly. All such charters held that the Governor and Selectmen (selected from among the share holders) would meet in London. The Massachusetts Bay Company charter did not specify where the meetings would be held. In 1630, the share holders chose a young lawyer from the Court of Wards and Liveries to be the governor, John Winthrop. Then he and the Selectmen (that's the origin of the term in New England local government) physically took the charter, took ship in Arabella
, and sailed to the new world to establish a "godly republic" in the wilderness. That's when Winthop delivered his "shining city on the hill speech" which Regan was to rip off 250 years later.
England suffered increasingly from strife between King Charles and Parliament, and in 1640, this dissolved into civil war. Massachusetts and Virginia were left to their own devices (and largely had been since James died in 1625), and both had to govern themselves, and to fight increasingly hostile aboriginal tribes. To do so, they found it necessary to trade with other nations in their West Indies colonies--chiefly France and Holland. Some few of the Virginians went back to fight with the King, and many Puritans from Massachusetts went back to fight for Parliament in the New Model Army. Charles had not time or attention to devote to the American colonies, and after he was executed in 1649, Parliament had two more years of war with Scotland, and the invasion of Ireland to occupy them. When Cromwell took over as Lord Protector, he ignored Virginia as unimportant, and Massachusetts Bay as already being "godly." Cromwell died in 1658, and in 1660, Charles II was restored to the thone. By then, the North American colonies had been on their own for more than a generation, and without governors, as well as without royal or Parliamentary interference.
Without going too much into the details, some colonies had very specific arrangements (Rhode Island had the right to choose their own governor, for example), or charters with wide-ranging powers (for example, Pennsylvania was virtually an independent "kingdom" of the Penn family). But basically, these established or nascent colonies in 1660 resented having royal governors appinted over them. They resented having restrictions put on their trade. The royal governors would appoint a council and attempt to govern without the local legislatures (such as Virginia's House of Burgesses), who in turn would starve the governor and council of money, since they knew their constituents would pay no taxes nor excises which had not been approved by the legislature. This dynamic became very important in the eventual formation of the American government.
In the Stamp Act crisis, committees of correspondence kept the various regions informed, and coordinated actions. After that crisis had passed, tensions mounted or abated, but gradually increased until in 1774, when the First Continental Congress was convened to deal with the now inevitable final crisis. Such a notional legislative body would have no governor nor council, already despised institutions. That congress was disolved when it seemed that tensions had abated again, but then the incident at Lexington, Massachusetts in April, 1775 lead to the establishment of the Second Continental Congress. This was the body which governed America during the revolution and afterward, until the constitution was ratified.
By the time the constitutional convention convened in Philadelphia in 1787, the institutions of royal and parliamentary goverment had been so long abandoned and had been so much despised a dozen years before, that there was never a probability of forming a government based on that model. The only state delegation which came to the convention with a proposal was Virginia. The Virginia plan called for a unicameral legislature, based on proportional representation by population, and a plural executive, which essentially would have been a committee of the legislature. This was one of the serious flaws of the Continental Congress and the Articles of Confederation--the President was appointed by Congress and had no independent authority. Some delegates from the small states (small by population, lead by New York and New Jersey--Virginia was the largest state by population then) had instructions not to approve a legislature based on proportional repesentation, and some even had instructions to withdraw if such a plan were even mooted. This was also one of the flaws of the Continental Congress--which each state having an equal vote, the more populous states would not support measures which would cost them more, and would simply refuse to cooperate if such measures were passed over their objections.
These boys were smart, experienced and well-educated. They resolved themselves into a committee of the whole, which allowed them to discuss anything they wished informally. This gave delegates time to contact their legislatures to get them to relax or suspend their instructions. The convention then addressed the flaws of the Continental Congress and the Articles of Confederation, and came up with a damned clever and a very flexible document to accomplish that. The Senate (equal representation for all states) has control of sovereignty (approval of executive appointments and ratifying treaties), and the House (proportional representation) holds the purse strings--money bills can only originate in the House.
As i've pointed out, there's a good deal of nuance left out. But basically, Americans were fed up with the English colonial model, and they were rebelling against the Parliament. It just never would have entered anyone's mind to have had a "parliamentary model," and the system they did come up with addressed the problems of the Continental Congress, and established a government with which Americans were already familiar, along lines of which they already approved.