But a coin will either be anonymous or transparent (or the option to choose one or the other per-transaction); no in-between. Is that not a correct statement?
No. For example, here are a few 100% transparent transactions. 
The 2 BTC on the left was mine. Now, which addresses on the right still belong to me? Nobody can tell.
Note that pretty much all wallets are HD nowadays. There is no mixing or other obfuscation trickery at play here, this is just normal default wallet behavior.
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Currently, our fiat transactions, aside from physical cash-to-hand, they are "kind-of" safeguarded by the centralized banking system, in-terms of some level of anonymity.
Hah, for the general public, maybe. They (banks, governments, secret services, paypal, credit card companies, etc) most definitely have all the information they want. All conveniently grouped together in a nice, efficient, centralized database. With fiat, your supposed precious 'privacy' is handed over on a silver plate.
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You can't walk into the bank and ask the teller to show you your neighbor's financial transactions of last week.
With cryptocurrency, you can't either.Quote
Convince me why I shouldn't be concerned about lack of anonymity.
I hope the above picture explains why this is completely infeasible. Still, if you're really paranoid, you might look at Monero's cryptonote protocol. That's 100% anonymity built in right there.Why doesnt Bitcoin Core work like that? (HD) it seems way more anonymous, all wallets should work like that. I use Bitcoin Core and i have no idea how to use the rest of wallets, and I dont even trust them to store real amounts of BTC.
FWIW, HD wallet is not more anonymous than the normal wallet.
TBH, I don't know what he really illustrated there and telling it as a default HD behaviour. Either he is splitting Bitcoins to new addresses or is sending changes to new addresses.
If you want to do either of the above, it is not mandatory to use an HD wallet. In all transactions, if you are sending an amount less than input amount plus fees, you need to give an address to receive change, else it will be used as transaction fee. So this is one of the basic thing of a wallet.
Bitcoin Core and all the known local wallets, use new addresses to receive change. If you want to split Bitcoins to multiple addresses, you can send your Bitcoins to multiple addresses. Its easy!