Note that I have very little understanding of blockchain crypto, so I am unable to confirm/deny the information OP gave. However, the way I understand the attack is:
Hardware wallet generates a private key. It keeps this key in internal storage. When a transaction is made, the wallet makes a signature. According to OP, there is a variable here (I'm guessing either multiple private keys, or the ability to choose a signature algorithm, or even embedding a timestamp in the transaction) which the hardware wallet can use to "leak" information.
Let's say the wallet decides to embed a timestamp. Whenever a bit in the private key is 0, the timestamp is even, and when a bit is 1, the timestamp is uneven.
After 4096 transactions, presumably the whole private key is now stored in the blockchain as even/uneven timestamps.
This is of course a very slow way of leaking the private key, but does illustrate the problem of having unverified devices be responsible for crypto results.