After 2,500+ words, the answer to the question is definitive:
If the profile picture is already visible to you (public or you are friends), here are three safe, free, and non-malicious ways to download it:
: High failure rate; often stops working after minor Facebook UI updates. Verdict
99.9% of dedicated "viewer" websites are scams. 0.1% are harmless but useless (they just show you the same small image you already see).