I don’t know if this is a joke thread, but I have no idea why it should be "patriotic" to win Olympic medals, let alone in a Singles’ sport. The Olympic Games is one of the biggest event of celebrating INTERNATIONALISM.
The idea is that athletes from around the world come in separated (that’s why they walk in under their country flags at the Openining Ceremony) and then become "friends" during the competitions and march out together and independent of countries at the Closing Ceremony. Of course that’s also a cliche and very few really embrace that idea, but definitely there is nothing nationalistic or patriotic about the Olympics.
Something like "Medal Count" is nothing but an artificial construct mainly promoted by some local media centers to give the casual viewers something to cheer for and therefore turn on the TV.
However, I love the Olympics and if I could choose one single title I want to win it would indeed be an Olympic Gold Medal, but I wouldn’t ever think of having won it for anyone else but myself then.