Why was it assumed that Kreacher follows the same rules as the enchantments on Grimmauld Place?

Question

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?

Answer 1

My understanding was that they were confident in the enchantments, if the house ownership had passed to Harry. There was a possibility that there was an enchantment in place that would mean Bellatrix inherited ownership, described slightly earlier in the chapter:

‘Black family tradition decreed that the house was handed down the direct line, to the next male with the name of Black. Sirius was the very last of the line as his younger brother, Regulus, predeceased him and both were childless. While his will makes it perfectly plain that he wants you to have the house, it is nevertheless possible that some spell or enchantment has been set upon the place to ensure that it cannot be owned by anyone other than a pure-blood.’

‘And if such an enchantment exists, then the ownership of the house is most likely to pass to the eldest of Sirius’s living relatives, which would mean his cousin, Bellatrix Lestrange.’

If Bellatrix owned the house, it wasn't certain if ownership would trump the protective enchantments.

As Kreacher was inherited along with the house, demonstrating that Harry was Kreacher's new master also settled the ownership question.

Answer 2

The concern isn't that the protections will recognize Bellatrix as the rightful owner - the concern is that ownership will give Bellatrix a backdoor that skips the protections.

Imagine your uncle buys a castle, and with it comes a massive keyring with keys to all the doors of the castle. Your uncle changes all the exterior locks he can find, and keeps the new keys separate. But there's a rumor that there's a secret passage under the castle, which can be opened by one of the keys on the keyring. Your uncle never identified all the keys on the keyring, so this is possible, but you have no way of verifying.

Your uncle dies, and the keyring is passed down to your worst enemy. Is it safe to live in the castle? They don't have access to the normal entrances, because your uncle changed the locks. But if the secret passage exists and your enemy finds it, they have free passage into your castle.

On the other hand, if the keyring passed to you instead, then it doesn't matter if the secret passage exists, because if it does only you can unlock it.

That is what the Kreacher test was for. It verified that Harry owned the keyring, so any theoretical secret passages through the protections were irrelevant.