For many of these cities, that the British named them so because they could not pronounce the names, is just an urban legend. For many of the renamed places, the current names are just as 'recent' historically as the British-given names are. In many cases, it was just politicians trying to capitalize on the anti-British sentiments and mislead people.
Historical publications from the 1700's and 1800's give very detailed accounts regarding the etymology for many places the British settled in. Definitely much more accurate and useful than the one-liners about self-respect and 'preserving culture' that our politicians keep throwing around now.
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Renaming Bombay, Calcutta and Madras was a rather brutal move. All three of these cities were completely built from scratch and developed by the British into the largest cities in the subcontinent. It would have been a nice gesture to retain these names in recognition of their contribution.
That said, like others have pointed out, this is not really an important thing when we have so many more important problems that need to be solved. But it is always a good idea for people to be aware of the exact history instead of just sticking to popular opinion, so that they do not form biased and potentially dangerous view points.

Not about their ease, but rather a limitation of the language and also with some influence from the North Indian dialects. And it is a limitation shared to some extent by North Indian languages as well.
For Bengaluru, Mysuru, Tumakuru etc, the main change was the vowel 'u' at the end of the name. The British did this with many places elsewhere too. Practically, many places in Andhra have names which end with a 'u' sound, but the Brits spelled them with an '-ore' at the end. Like Nellore, is pronounced 'Nelluru' in Telugu, but written as 'Nellore'. Similarly Guntur (GNT) is pronounced 'Gunturu'.
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Most of these names based on South Indian dialects do not fit well conversationally in English, so they were spelled without the vowel sound at the end. A typical south Indian would still read 'Bangalore' as 'Bangaluru' or Mysore as 'Mysuru' as this is the natural slang in south Indian languages.
Even to this day, a lot of North Indians casually refer to Kannada as 'Kannad' or Kerala as 'Keral' as ending words with a vowel sound is far more common in Telugu/Kannada dialects than any other language in the country.
IMHO, these were just minor changes trying to 'Devanagari'-fy the English spelling. However they spell it, south Indians are always going to use the 'u' sound at the end, and foreigners/other Indians will continue to cut the vowel sound at the end.