Does not writing tests actually save time?

One of the most common responses to why people don't write tests is, "We don't have time".

So, people don't write tests to save time.

While you write less code, it takes longer to test everything manually during development, and more bugs will likely be found, which also takes longer to diagnose, fix, and deal with any knock-on effects.

Bugs in production add more administrative overhead as new tickets need to be created and managed, prevent other tickets from being worked on whilst they're fixed, and require more communication with clients and customers.

If you save ten minutes not writing a test but spend hours later fixing bugs, you didn't save time - you spent more time in the end.