I think you're thinking of Loyalists, Federalists, and Anti-Federalist Republicans as being all from the same era, which they were not. Loyalists were strictly around and vocal during the years leading up to and the years throughout the American Revolution. By the time it was over, many either had or would leave the newly established United States for Great Britain or another colony, most popular was parts of eastern Canada. The establishment of Federalists/Anti-Federalists did not come about until years after the war concluded. Both sides were active in the early years of the Republic, especially during Washington's second term in office, with Federalists backing such figures as John Adams and Alexander Hamilton and Anti-Federalist, Democratic-Republicans, backing figures like Thomas Jefferson and James Madison. Loyalists played no role because they had essentially backed the wrong horse. Whether Federalist or Democratic Republican, people who made up these parties on both sides were either patriot or neutral (it's estimated that 40% - 50% of the American public were neutral during the war) during the war.

But if you're asking why did Loyalists who remain in the United States not try and be more active in American politics? Well, the reason is rather simple. Loyalists, especially members of the gentry who would have been in a position to attain political power, were largely disrespectful to the American masses. As award-winning historian Alan Taylor put it:

The newly aggressive politics of common men troubled many gentlemen. Gouverneur Morris of New York worried, “The mob begins to think and to reason. . . . I see it with fear and trembling, that if the disputes with Britain continue, we shall be under the domination of a riotous mob.” He concluded that the “mobility grow dangerous to the gentry, and how to keep them down is the question.” The revolution divided gentlemen, with the most conservative becoming Loyalists who defended British rule as essential to social stability. But Patriot gentlemen felt more confident that they could manage the common people. James Duane advised, “Let [the people] be rather followed than driven.” Cadwallader Colden noted that “gentlemen of property” took charge of Patriot committees lest “the most dangerous men among us would take the lead.” While Loyalist leaders dreaded the politicized common people, genteel Patriots hoped to control them.

Both Patriots and Loyalists were cross-class coalitions led by rival gentlemen who rallied common supporters. In the northern colonies the wealthy divided, but the richest southerners overwhelmingly became Patriots (or “Whigs”). “There were in no part of America more determined Whigs than the opulent slaveholders in Virginia, the Carolinas, and Georgia,” recalled David Ramsay of South Carolina. Where the gentlemen of a county united, for one side or the other, they drew almost all of their neighbors with them. But in many counties, gentlemen divided into rival camps and competed for popular support...Loyalists claimed to defend true American liberty, for they regarded Patriot committees and mobs as greater threats to freedom than the small taxes levied by Parliament. (1)

So you can see, Loyalists sort of shot themselves in their own collective feet during the American Revolution and remained just as loathed after the fighting stopped. This happened even after the Treaty of Paris guaranteed some protections to the Loyalists, yet many if not most Loyalists did not find the protections that it guaranteed.

So why didn't Loyalists find a place for themselves in the new government? They were alienated from start, with many fleeing the country in order to avoid further repercussions for their politics during the war.

1). Alan Taylor. Alan Taylor, American Revolutions: A Continental History 1750 - 1806 W. W. Norton & Company. 2016. pp 121 - 122