Chapter 3 of Half-Blood Prince makes it very clear that The Order doesn't quite knows how the enchantments on Grimmauld Place work:
We do not know whether the enchantments we ourselves have placed upon it, for example, making it Unplottable, will hold now that ownership has passed from Sirius’s hands. It might be that Bellatrix will arrive on the doorstep at any moment.
This problem is resolved by testing if Harry owns Kreacher ("if you have indeed inherited the house, you have also inherited [...] Kreacher". But why does this resolve anything to do with the enchantments? At best, the Kreacher test only proves that in the eyes of the magic that binds Kreacher, Harry owns the house. This says nothing about the magic that's on the house itself. Furthermore, we already know that the magic that binds Kreacher isn't foolproof, as pointed out by Sirius' confusion regarding Kreacher not following Tonks' orders and on top of that, Deathy Hallows repeatedly shows elf magic breaking wizarding magic's rules. So why should we believe that Harry's ownership of Kreacher is the solution to this problem?