Remember: It is not important how the tip feels; it is important how it looks. The very nature of a point creates some bulk in that area; the sharper the point the bulkier it will feel. Bulk is not minimized by cutting away fabric; that only weakens the point. Bulk is minimized by sweeping the edge of the fabric away from the tip.
Note: For these photos we used silk organza fabric so you can easily see what happens under the applique, and red stitching thread.
Here is how we create pointy-points: