-ing clause vs having + past participle

Thank you, Abdul, for such a well-documented question.

@Abdul Rahman posted:

I came across this sentence in 'English Grammar in Use':

_______a hotel, we looked for somewhere to eat. [Finding - After finding -  Having found - We found] and the answer key of the book is 'After finding & Having found', which I completely agree with. Yet, why not 'Finding' as well.

Is it because as is mentioned in some grammar books to avoid ambiguity and to indicate that 'Finding a hotel' took place first, then it was followed (not immediately) by the second action, 'going somewhere to eat'?

Exactly. "Finding" does not work for the reasons you mentioned. See how these work:

- Looking for a hotel, we found somewhere to eat. (As we were in the process of looking for a hotel, we found...)

- Seeing that there was no hotel available, we decided to find somewhere to eat. (As soon as we realized our search for a hotel was to no avail, we decided ...)