Britain has a rich and surely fascinating history. But becoming a British citizen basically boils down to voluntarily becoming a subject of the British crown. All you need at that point is a primer on how to turn the lights on and off.

America, in contrast, is an ongoing social and philosophical experiment more directly defined by its immigrant population than its native-born population. You enter America more as a collaborator in the construction-of-America project than as a subject of her constantly-shifting government. Therefore you need the historical and philosophical background.

by Almighty God that, on becoming a British citizen, I will be faithful and bear true allegiance to Her Majesty Queen Elizabeth the Second, Her Heirs and Successors according to law.

That's ridiculous. There have been no crown subjects for decades - when you become a citizen of the United Kingdom you're becoming a citizen of a liberal democratic state, just as you are when you become a US citizen.

Every country in the world is a social experiment, as can be seen by their ever-shifting constitutions and laws.

There have been no crown subjects for decades

On 1 January 1983, upon the coming into force of the British Nationality Act 1981, every Citizen of the United Kingdom and Colonies became either a British Citizen, British Dependent Territories Citizen or British Overseas Citizen.

The use of the term "British subject" was discontinued for all persons who fell into these categories, or who had a national citizenship of any other part of the Commonwealth. The category of "British subjects" now includes only those people formerly known as "British subjects without citizenship", and no other. In statutes passed before 1 January 1983, however, references to "British subjects" continue to be read as if they referred to "Commonwealth citizens".

British Citizens are not British Subjects under the 1981 Act. The only circumstance where a person may be both a British Subject and British citizen simultaneously is a case where a British Subject connected with Ireland (s. 31 of the 1981 Act) acquires British citizenship by naturalisation or registration. In this case only, British Subject status is not lost upon acquiring British citizenship.

The status of British Subject cannot now be transmitted by descent, and will become extinct when all existing British Subjects are dead.

British Subjects, other than by those who obtained their status by virtue of a connection to the Republic of Ireland prior to 1949, automatically lose their British Subject status on acquiring any other nationality, including British citizenship, under section 35 of the British Nationality Act 1981.
...

Although the term "British subject" now has a very restrictive statutory definition in the United Kingdom, and it would therefore be incorrect to describe a British citizen as a British subject, the concept of a "subject" is still recognised by the law, and the terms "the Queen's subjects", "Her Majesty's subjects", etc., continue to be used in British legal discourse.[7]

I still stand by my theory that a basic grounding in history, culture and philosophy is what you would want in a "citizen", whereas you would likely focus on the practical for a "subject."