Thanks, now I don't feel like the most paranoid person in the room. There are risks with any approach, but buying a physical gift card for cash still seems to be among the safest of all the alternatives.The recipient does not, but retail chains use face recognition to look for known thieves and track customer behaviour. Amazon itself already offers Amazon Rekognition for free as a promo.
Facial recognition by a retailer does not give them much - they'd need to cross-check the person with the specific gift card number. Even if they can track it, they don't know where it was sent and who claimed it. It would take effort and a combined data from the retailer and Amazon (for Amazon cards) to figure it out. It might happen with a court order during a high-profile investigation, and that's about it. Unless the Big Brother has it covered, in which case we are screwed anyway.
Amazon eGift card provides convenience at a cost of a bit more risk. It has paper trail within one company - a single customer support representative at Amazon could probably connect people who claimed the gift card and who bought it (if it was bought with a real name / email address / credit card). I could accept this small risk, or could get more tricky by creating a separate Amazon account under an alias for buying eGift cards and fund it by a yet another paid-for-cash prepaid card or Amazon card.
Gift cards for smaller amounts won't raise an eyebrow. Purchasing gift cards for large amounts should attract more scrutiny than using cash. Whatever makes gift cards great for anonymous transfers, also makes them ideal for money laundering and fraud. From The Guardian article "Attempts to stay anonymous on the web will only put the NSA on your trail": "Her husband tried to buy $500 of Amazon gift vouchers with cash, only to discover that this triggered a warning: retailers have to report people buying large numbers of gift vouchers with cash because, well, you know, they're obviously money launderers."