What I've noticed is that in both the internet as well as in books, that both systems of translating Mandarin Chinese into English (Pinyin and Wade-Gyles) are used and I'm curious as to why.
(Note, for both translations, the bold is the one I've seen commonly used and the one in brackets is the alternative).
Some examples for Pinyin translations I've seen commonly used (I'm using two tables that give both and using any that I can easily recognise. ):
And some examples for Wade-Giles translations I've seen commonly used:
Other things to note:
Any patterns and/or trends I've noticed:
Anything else:
If anyone has an answer, I'm curious to see what you knowledge, insight, speculation, thoughts, etc. are. And thank you if you read all of this post.
A past answer of mine discusses some aspects, although it is worth noting that four of your five examples listed as Wade-Giles are in fact not Wade-Giles at all. Whampoa derives from Cantonese Wong bou rather than Mandarin Huangpu, and this Cantonese-derived rendering was preserved as the 'Postal Romanisation' for the city. Chiang Kai-Shek and Sun Yat-Sen are similarly Romanisations of Cantonese, not of Mandarin. Yangtze comes via the 18th century French rendering 'Yang-tse' – in Wade-Giles it would be Yang-tzu.
This, I think, gives a bit of a clue to your question. Pinyin has basically been the standard in academic writing since 2000, with a handful of exceptions such as entries in long-running series like the Cambridge History of China (although even then, the latest volume has switched to Pinyin). However, a large number of customary Romanisations, not based on any formal standard (Wade-Giles, Yale or otherwise), have remained in use either out of convenience or indeed appropriateness, as the case may be. For instance, Taiwan retained the use of Wade-Giles until 2002 when it switched to Tongyong Pinyin, and only formally switched to Hanyu Pinyin in 2009, but even then places have tended to use the customary Postal Romanisation (Taipei, Kaohsiung, Tamsui etc.), while many people and organisations have opted to continue to use the earlier Romanisations, hence a number of Taiwanese historians Romanise their names using Wade-Giles. And, as noted, Cantonese is its own language and so things like local Hong Kong histories often opt to transliterate the Cantonese readings of names – though not all historians do so.
EDIT: Although I try not to be the pedant as much as I used to, there are a few things I did want to flag because they may have caused you to jump to certain conclusions. 'Peking' was the Postal Romanisation, while 'Pei-king' was virtually never used; Whampoa is a city in Guangdong, and the site of the Whampoa Military Academy (hence, a place that is not rendered in Pinyin); 'Yangzi' is the Pinyin for the river rather than 'Yangtzi'.
And entirely separately, 'Pu I' is technically how Puyi's name would have been rendered in Manchu. It's unlikely that was intended by whoever you were reading of course, but it's a remote possibility.