TR would have had to have had a stroke and gone insane for this to happen. What possible source of conflict is there between Britain and the US by WWI?
As 1940LaSalle said in the following post, TR in OTL was indeed an anglophile of long standing, but that merely means that the POD would have to be pretty early in his life - though the resulting butterflies would not be visible beyond his immediate personal circle for many years.
My biographical knowledge of TR is only casual, and I can't say under what specific circumstances he might have acquired anti-British sentiments in youth instead of pro-British ones, or how that might have affected his rise in public life. But I can't see anything inherently ASB-ish about a POD that begins with TR having different attitudes.
As to your second and broader point - from at least the 1890s on there were plenty of potential sources of conflict. The US, like Germany, was a rising industrial power, implicitly challenging British exports in world markets. The US, like Germany, was also a rising Great Power, seeking a place in the global sun, also an implicit challenge to British hegemony. The US, like Germany, was specifically building a very large navy, far larger than it needed for direct self-defense against any force other than the RN. If the US had been anti-British, it would have posed as serious a problem as Germany did in OTL - in some ways worse, because the US would be harder to blockade and had a potential hostage in Canada.
Look at it in terms of an alternate-alternate history: How stupid was it for
Germany to antagonize Britain?
Germany's strategic threats in the late 19th and early 20th century were France and Russia - the same as Britain's strategic threats. By all logic, Germany and Britain should have been if not allies at least friendly neutrals toward each other. Suppose that the buildup of the German navy had not been accompanied by pervasive anti-British rhetoric - would the tensions that led to British entry into the war ever have developed?
I'm simply postulating that rhetoric and psychology matter, and that in the right (or wrong) conditions the Anglo-American well could have been poisoned just as the Anglo-German one was in OTL.
Remember that Anglo-American relations even in the later 19th century were correct but by no means warm. The Revolution and 1812 were closer to living memory, Britain was still widely perceived as a squirearchy rather than an emerging democracy - and one very large segment of American public opinion, Irish-Americans, was deeply anti-British.
One important thing to remember: British diplomacy was excellent, and would do whatever it took to keep the USA out of the war. In addition, if tensions started building towards that point, and the USA started gearing up, the British and French would both know that an American entry would be catastrophic. Just the loss of Canadian food and troops would be crippling.
I can't see any practical way for the USA to project power into Europe in the face of British naval superiority and a lack of bases. (The USN has modern battleships, but minimal cruiser resources for commerce interdiction and protection in distant oceans.)
The first point is true, but in an age of nationalism, diplomacy could only go so far if anti-British sentiment in the US was being whipped up.
On the second point, the US wouldn't have to project power directly into Europe - as you point out, it could inflict crippling damage on Britain without a single doughboy setting foot in Europe.
In the pre-dreadnought years the US had a formidable cruiser force, including the "Big Ten" armored cruisers - individually at least comparable to Spee's cruisers, and with a far better operating situation. In OTL the US never completed any battlecruisers till the
Alaskas, but in an ATL where anti-British attitudes are in the ascendent, it could easily have built some.
-- Rick