I’ve had super unresponsive fixed axles before. And it’s strange for sure. You can bind… Or double loop… But yeah, if the gap can be adjusted, that’s what you’ll want to do first. If not, maybe try a little wax or Vaseline around the tip of the string. (A micro micro amount, maybe half the size of a match head)
And… Try harder tugs to bring it back up. Wood is also just finicky and semi-unpredictable. Part of the beauty of the beast!