With the exception of the Revolution and the War of 1812, the US and UK never really were adversaries – even "rivals" would be a misnomer.

I know you're more interested in 1812-1917, but looking at the American Revolutionary War is important for the relationship between the two countries. The American Revolutionary War was viewed at the time not as the romantic birth of a nation but a British civil war where neither side achieved their aims and ultimately agreed to leave each other be. Obviously, the US "lost" less than Great Britain, but even the Declaration of Independence was Plan B.

The British aims during the Revolutionary War were obvious – put down the rebellion – but the Revolutionaries' Plan A was to reform the British Empire. The colonies believed they were fighting for their "rights as Englishmen" and that Americans deserved seats in the British Parliament if they are to be taxed by that very legislature. There were certainly many Loyalists who sympathised with these aims but considered the Revolutionaries to be terrorists, and after 1776, many fence-sitters stopped musing these sympathies as the Declaration of Independence made loyalty to Empire an issue. Woodrow Wilson reckoned that Loyalist forces actually outnumbered the Patriots in 1779, but that claim is dubious.

Of course, George III and the British Parliament did not acquiesce to American demands, so the Founding Fathers cut their losses; if we cannot better England, then we shall make our own England.—hence 1776. It is worth noting that the first flag of the United States had the Union Jack.

Fast forward to the leadup to the War of 1812. Britain and France had been policing the Atlantic ocean during the Napoleonic Wars, to the dismay of American merchant sailors who – neutral in the conflict – were often seized by the Royal Navy and Marine Impériale for being en route to their enemy's ports (but seized more often by the former due to naval superiority; 15,000 American sailors were kidnapped by the Royal Navy from 1793-1812, even capturing them in American waters). The US imposed an embargo on all foreign trade in 1807 to coerce Britain and France into behaving by depriving them of highly sought American goods – this was a disastrous policy:

• Britain's naval dominance meant this would have little effect on the French economy which the Royal Navy had already blockaded

• Britain was able to find new markets in Latin America

• Britain did not have to compete with the United States in these new markets

• The United States' GDP suffered a recession of 4.1% over the next year (similar to the 21st century's Great Recession)

The US lifted all her embargoes in 1810, but American sailors were still undergoing British impressment, and exports to continental Europe were still off the table. In order to have a sovereign trade policy, many in Washington felt a won war would put Britain in her place. However, whereas every other Congressional declaration of war has been almost unopposed, the 1812 declaration of war against Britain was not a unanimous decision; 38.3% voted nay in the House of Representatives and 40.6% nay in the Senate. It was also unpopular with the mercantile class who enjoyed lucrative trade with what was then America's primary trading partner. By the time Napoleon was exiled to St Elba, the initial premise of the war was no longer relevant, but both sides didn't want to leave without any accomplishments; the British would famously go on to burn down Washington DC and the Americans would famously go on to hold bay at New Orleans.

But aside from these two wars? The two countries had a glowing trade & cultural relationship, AND were on opposite hemispheres; almost any disagreement would be the other party's business:

• The US and UK were each other's primary trade partner for the first half of 19th century

• The United States was still receiving plenty of immigration from Great Britain; in 1920, 1.3 million Americans were British migrants, 2.3 million were the children of British migrants, and 3.7 million were the grandchildren of British migrants

• Old Stock British Americans were still the main political and cultural force in the United States, who numbered 31.8 million in 1920

• Even as recently as 1980, 50 million Americans identified as English on the census

• America and the Empire were and still are importing each other's culture; Jane Austen was probably more revered in New England than Old England.

They could trust each other to do business with one another and keep each other's adversaries out of each other's hemispheres. Come WWI, the US begins preparing for war in 1915 after the Lusitania attack, and come 1917, with domestic turmoil in Russia, and Germany escalating the conflict through the Zimmerman telegram and resuming unrestricted submarine warfare, President Wilson asked Congress to help out America's biggest debtor.

TL;DR Anglophilia because Americans considered themselves Englishmen and the two countries had important trade links with each other. The Revolutionary War (both sides lost) was fought over imperial reform and the War of 1812 (neither side lost) was fought over existing economic warfare spurred on by the existing Napoleonic Wars.