> The Mughals eventually came to see themselves as "Indians"(sic) and built their homes here.

That is the key difference. The British were never interested in anything but looting India. The Mughals were more like the Normans in British history -- they started off as conquerors but eventually completely assimilated into the local population.

For those who read Hindi/Urdu, this ghazal, which eventually became his ephitaph, by the last mughal emperor Bahadur Shah Zafar is quite touching (https://rekhta.org/ghazals/lagtaa-nahiin-hai-dil-miraa-ujde-...). The whole ghazal is well worth reading, but it ends with:

  कितना है बद-नसीब 'ज़फ़र' दफ़्न के लिए 
  kitna hai badnaseeb 'zafar' dafn ke liye
  दो गज़ ज़मीन भी न मिली कू-ए-यार में 
  do gaz zameen bhi na milee koo-e-yaar mein

Rough translation: How unlucky is Zafar; he couldn't even/be buried in the streets of his friends.

(The British exiled Bahadur Shah Zafar to Rangoon and he was eventually buried there.)