It seems that Facebook's applications for generating Instagram tokens do not always work correctly. After trying multiple methods without success, I decided to inspect the blank page where the token should appear.
How I Found the Token:
Open Developer Tools (Inspect) in your browser (F12 or right-click > Inspect).
Search for IGAA in the page's source code (Ctrl + F).
Alternatively, go to the Network tab and look at the response of the first request after attempting to generate the token. The token should appear there.
Even though the page might be blank, the token is sometimes still present in the response.
This method worked for me when the standard process failed. Hopefully, it helps others facing the same issue!