Users with private profiles and possible unactivated wins

Make a support ticket and ask them to check. Or write message to winner and ask to change.

a) SteamGift Tools is not official. Is was created by a user but it does have the official blessing by cg that it's ok to use it to protect your GAs.
b) I don't think that's technically possible. Honestly no idea how mods can check without asking that person to switch their profile to public. Maybe they are using logs from the last time that user synched their account.

I agree private profiles are annoying. It's not like anybody forces you to post your social security number on your profile.

For now when you get a winner with a private profile simply file a reroll request and add in the description that the winner has a private profile and ask if Support would be so kind to check for unactivated and multiple wins. If they find anything that warrants a reroll they will automatically do so.

You're within your rights to ask that the winner make his account public for a bit so that you can check it before giving the prize.

But, as much as I support users' right to a private profile, that's still quite an inconvenience. SG does keep a list of every game on your account from your last sync. so it's technically possible for cg to make a tool to help winners check. It could make note of games marked "received" more recently than the latest sync to avoid false negatives, and indicate or omit games that the winner has already received a suspension for, to cut down on unnecessary re-roll requests.

Users with private profiles

You can ask SG support to check those and either they will say that the account is in good order and you can send the gift or you get a re-roll.

and possible unactivated wins

You can ask this also to SG support or you can check them on SGTools (https://www.sgtools.info/activation) and ask for a re-roll if your winner does have non activated wins. You'll get a response either that the winner already served punishment or you'll get the re-roll.

SG should automatically check winners accounts. If SG cannot sync a users account because it is private, they should get a popup letting them know and not be able to receive any wins until they set their profile to public and manually sync with the site. They can set their profile back to private right after. If they refuse to sync their account within 7 days, they forfeit their win and it is automatically rerolled.

From my previous comment here, I think SG should check every winner before sending the game and automatically suspend rule breakers.

I think it should be a rule to have public profile to use sg.