Why a dropped index is still there, and does it prevent a table from being altered?
Why the dropped index is still there?


In fact that index is no longer there, the column is the one still there and what was once a Clustered Index is now a Heap.


Did I got the dependency error above because of this still-existing index?


No, because the index no longer exists.

If not, is there something else I should check? I'd like to avoid
dropping the table.

You can View the Dependencies of a Table as suggested by Erik Darling.

As you mentioned in your comments, you found a VIEW using this method and if you check the CREATE VIEW doc, you'll see the option SCHEMABINDING could cause the error you mentioned:

SCHEMABINDING Binds the view to the schema of the underlying table or
tables. When SCHEMABINDING is specified, the base table or tables
cannot be modified in a way that would affect the view definition. The
view definition itself must first be modified or dropped to remove
dependencies on the table that is to be modified. [...]