IMO, one of the most time-saving use cases of LLMs is handling the syntax of a new library, which can be time-consuming to learn if it's only needed, say, once. They can also help save time when finding bugs in code, especially when the bugs are common.
Mehmet Mars Seven ♞
LLM's speed up writing code that scientists use in everyday work by a factor of at least 3 if you know how to use them well
Ben Golub
The closest thing to specific example of a use case I’ve heard is one of the hosts of the Acquired podcast saying—in a paid ad for Claude—that Claude helped him quickly make a table of dates from his research notes.