Okay; so because I didn't buy via KYC exchange in the first place (therefore 'not traceable in my name'), they'll ask more questions compared to another sale and may / may not steal my fiat money?
One of the things they check is indeed if you sent money to a centralized exchange In the past.
Then I guess a P2P (cash in hand) trade of those BTC might be possible or otherwise a direct sale for BTC. In that case you will be sure not to get anything stolen from you and as long as your tax documents are fine, there won't be issues on that front either.
Unless you decide you want to deposit the cash (in a large amount) to your bank again.
That's because the bank has liabilities / duties to the country it operates in and a registered exchange (company) is an easy / no headache source of funds for them, right?
The banks just tries to cover their asses, especially after the huge fines they've had recently for not doing enough against money laundering. Even though, as far as I know, those fines were based on fiat money laundering.