In which cases do you request a new winner?

You wait 7 days if he doesent leave feedback report to support

You need to wait 7 days. Try to contact him (add on Steam / send an email) and only then open a support ticket.

You already long time on Steamgifts, maybe now its time for read FAQ. =)

If you gave game away, what good requesting new winner would do?

you need to wait a week before asking for a new winner, unless they already have the game and entered on accident or something along those lines happen, them not marking the game as received right away is pretty common here ive had people wait til the last possible minute to mark there game received.

AFAIU he received the game but didn't tick the received. If so, try to contact him by Steam add (might need info box on your profile explaining why), email (can fall on spam) or most likely a comment on one of his old GAs so that he is warned about the situation.