Coming across this handy CSS Cursor tool I wondered: why are the CSS cursor types other than
text so rare on the web? Even most web applications don't go further than these three plus perhaps cursors for moving and grabbing.
I think you've answered your own question.
The special cursors demonstrated on that web site are rarely needed, whether in a browser or outside of one. Of the 31 cursors, 14 of them are for resizing elements, which isn't really a common task.
I see Progress and Help used fairly regularly.
Other than that, the rest of them are mostly situational... there's no need to use them out of specific tasks and environments.
Using cursors where not absolutely necessary violates the rule of don't confuse your users, ever. If you can use a normal cursor, do.
"Default", "Pointer" and "Text" are defaults in browsers.
Others we forget to specify for developer — because we paint static images. But if we work with interaction our-self, we will remember to use "Not available" cursor for disabled elements for example.