If your database has proper RI defined for it then there shouldn't be any case of compromised data integrity. All of your related tables should have declarative RI, which means that you can't delete a parent while it still has children.
Also, if you have code that is only deleting some of the rows at times then that is poor coding and poor testing. These kinds of actions should be a single transaction. Your suggestion of using a stored procedure is a great approach for solving that problem and is pretty standard.
As has already been mentioned, cascading triggers have the danger of deleting rows that someone did not intend to delete. Consider that sometimes people might be accessing your data from somewhere outside of your application, especially when fixing data issues. When someone accidentally tries to delete the wrong parent and gets an RI error that's good. When they accidentally try to delete the wrong parent and it not only deletes that parent but 20 children in 5 other tables, that's bad.
Also, cascading deletes are very hidden. If a developer is coding a delete for the parent then they should know that they have to use the delete stored procedure to take care of children. It's much preferable to have a developer not code against that, get an error, and fix his code (or realize that he doesn't really want to do all of that deleting) than it is to have a developer throw in a delete and have no one realize that it's killing off children until the code has gone live.
IMO, I prefer to have my developers knowledgeable about the application rather than make it easier for them to remain ignorant of it.