Do I miss something here? Why should I use HD wallets if I still need to store each single local private key as well?
So it seems like you've misunderstood this part:
Each address derived by an HD wallet looks just like any other bitcoin address, there is no way to know it came from an HD wallet. So in order to spend the UTXO, you need to present a signature made by the private key of that specific address. The network does not know that your address was derived from an xrpiv key, so if you use your xpriv key to sign the transaction, it will be invalid.
The comment is based on the OP's 3rd question which is: "
Do we use master private key to sign every transaction?" he actually nailed it.
To put it simple, as you know it (
Hierarchical Deterministic) HD Wallet has a Master Private key where all of your addresses's private keys were based from.
The client (
ex. Bitcoin Core/Electrum/Multibit) where it was created or supported can restore all those required private keys to spend the previous UTXO, your client does that automatically for you.
The only backup you need is the Master Private or the SEED, each private keys are only optional.
Also, I think that there are more information provided in the comments from the link you provided (
small fonts with numbers), look again.