I think this question requires a bigger overview than simply looking at how Victorian Britons looked at their empire. The historiographical understanding of how European empires were built is heavily coloured by a few things that dominate the discourse.

One such historical trend is the so-called Scramble for Africa in the mid 19th century, in which it is fair to ascribe a motive to governments of the day to accumulate colonies in Africa before competitors ‘got the goods’. The other I am thinking of is the Hispanic conquering of the Americas, and the way Europeans in all parts of the Americas subsumed the geographical presence of native peoples. These concepts loom large in our understanding of imperialism, because they underpin many issues that feel directly relevant to modern Americans, both north and south, and to modern Europeans. Europeans have to engage with many parts of Africa’s troubled experience of the 20th century through the lens of their countries’ roles in shaping its modern nations and the degree of success their political structures and processes have had in facing challenges. Americans face the fact that they live on land that they took forcibly in order to live there. Both have to deal with the facilitation of the institution of slavery.

Britain’s role in India began differently. The British did not arrive in India seeking to own it or to impose a British system of politics or legalism. Independent traders, not government-sponsored missions, arrived to trade with sophisticated markets that were already developed.

Niall Ferguson notes the irony that the British developed Indian connections because they were late to the game in Asia. Other European countries already had firm trade empires developing, like the Dutch in Indonesia, the Spanish with American gold, the French in South East Asia, with spice trade being the most valuable in Asia before industrialisation. India’s chief benefit was in the trade of textile material, and this was massively important when Britain became the first nation in the world to undergo industrialisation. They were able to treat India as an almost unlimited source of raw material for their milling industry back in Britain. They then had an enormous market to sell the finished goods back to in India. You could not sell the product of a diamond jeweller back to native South Africans who mined the diamonds, because they could not afford it or did not desire it. By the mid 19th century this protoglobalised relationship underpinned so much of the British economy that the British government could not risk its collapse, and in 1857 when Indian soldiers employed by the East India Company mutinied they took direct crown control of the administration of India to ensure it was governed in a sustainable way, as it was too big an endeavour to fail - a bailout of its day, in a way.

By this point, India was not merely a trading partner. Just as the relationship underpinned British economies, so it sustained the state coffers of many a prince, sultan, nizam, or maharajah. Further, the East India Company had been aggressive in its insertion to the subcontinent’s markets, and had pursued a policy of taking control of states where the ruler had no heir, or would sponsor one party in a feud between two states under the proviso of territorial spoils. Thus, after 1857, much of India was directly controlled by the British government, and the rest was fragmented and lacked unity - ‘India’ as a nation was not a precolonial concept. India was a region comprised of many protonational states. These states now owed allegiance to the British, and the British had the power to face down any dissenting ruler who thought they had the power to stand alone - and alone they stood, because the British doctrine effectively divided and conquered by playing the states against each other.

All this is well and good, and the value of India’s markets might be sufficient to answer your question. But I do think there is more to it. As a colonial possession, India comprised developed political infrastructures that could sustain investment that increased returns for British sponsors and industrialists in a way that less developed economies in Africa could not. For example investing in railways was lucrative in India but a risky venture in Africa with no native middle classes to ride them in great number. The sophisticated political frameworks inherited by the British included armed forces, in huge numbers, which could be deployed around the world as needed without risking politically costly British lives. Indian troops were used to put down African rebellions in several instances. And the army’s costs were paid for by taxes raised in India - this was enormously valuable to a small island nation. The Royal Navy paid for itself several times over by ensuring the connections between India and Britain went unthreatened.

Finally, I also think that the way the British conceived of this imperial possession plays a big part in its importance. They did not merely own it and give it a British name, like New Hampshire or something twee like it (sorry Jed Bartlett), they embraced India’s exotic ancient otherness and were dazzled by it. No other British colony was worth appointing a viceroy - a literal vice king. No other colony accorded the British monarch the title of Empress or Emperor. The history of India was not unknown to Europeans, and to have power over it was like owning the Eastern version of the Roman Empire in its years of decline. British universities appointed chaired professors of Sanskrit in this era for the first time, counting it as one of the classical languages alongside Latin and Greek. They treated Indian princes to state dinners, and British aristocrats hunted on their estates. By contrast, in North America the British made treaties to occupy land for white colonists. Africa was carved up into unnatural square-bordered sections that were put to work for raw materials, but no interest was really found in exploring the precolonial cultures of these possessions. In India, the British worked even to preserve the otherness of India, with the viceroy holding parades in which each state participated in a way that showed their distinctive qualities, with demonstrations of regional garb and martial practice.

So I feel that in these ways Britain’s relationship with India was unlike any other imperial relationship enjoyed by 19th century European empires, and its unique qualities came to represent the legitimisation of the entire imperial mission. Without India, the British Empire devolved quite rapidly. I don’t want to start another topic, and my last remark risks being a transhistorical generalisation, but I do want to mention that truth to underline the points I have made. Hope this answers a few things for you.