When the kids in our family were little, we wrote the notes to the other kids, or kids and parents (depending on who really gave the gift), because we felt it was important for kids to get the sense of receiving a thank you and then get in the habit of sending them themselves. We wrote them when they were very little, and wrote in our children's "voices" as if they really spoke, but by 3 we would just write down what they said about the gift (if it was a toy) or whether they thought it was "cool" (if it was clothing). If it was a gift card, then we just wrote that we would go there and pick something out. And we'd fill in the rest of the note around the child's own comments. After the kids can write a little, we sometimes did a 'fill in the blank' thank you note so at least the kids participated. If your daughter draws a circle or a squiggle on the notes, I think that's charming! Maybe she can put the stamp on too, or put a sticker on the flap after you seal it - anything is a good exercise in manners.
You aren't getting thank you notes sometimes because people don't know enough to send them - same as they don't RSVP! Pet peeve! So by all means, take the time to show your appreciation and help spread the idea of gratitude! People who don't feel appreciated don't keep coming to parties and don't give gifts. We cut people off after a certain amount of time if we never hear from them.
I just saw on line with my bank that a kid we've known since kindergarten deposited the check we gave him for high school graduation 4 weeks ago, but we don't have a thank you yet. I'm annoyed as hell. My son wasn't allowed to deposit the check or use the gift or gift card until the thank you was written.