Analyzing 'genitive/accusative + V-ing phrase (gerund-participle phrase)' as different constructions

I don't think the approach shown in CGEL is problematic. Basically, their internal structures are the same and there is no syntactic difference between 1 and 2. Choosing accusative or genitive is a matter of style. A syntactic analysis does not have to show you the semantic difference and I don't see any semantic difference between 1 and 2. If somebody claims that there is semantic difference, the syntactic analysis does not have to deal with this

Your suggested approach is more problematic. If "him" is the object of regret, you are saying that you regret him. But what you regret is the fact that he left. You don't regret him.