I might ask why the no wrapping request is made. Some people are very shy and would be mortified by sitting in front of so many people. Sometimes there is a major discrepancy bw the families' (in laws) incomes and opening in front of others would highlight this. There may be a cultural reason or they just may be environmentally concerned. I find that asking works better than pondering. I also prefer to assume good motives. Maybe the recipient wants to focus more on the people and less on the gifts.
I actually wrap all birthday gifts in newspaper and skip the card. I write directly on the paper. Shredded paper works well instead of tissue paper when I reuse the gift bags I receive.