Flagging just means the money laundering team would investigate the source, determine that it doesn’t have other red flags, and quietly move on.

Exactly. Staying with OP’s hypothetical situation, potential investigation on its own does not mean it can’t happen. It just means there may be an extra step behind the scenes, very possibly without either party’s knowledge anything even happened. More importantly, if the benefactor insists on some dodgy payment method that’s an even clearer sign of BS because a normal bank doesn’t give as much of a shit as OP lets on