Holding mince under a pinch doesn't directly make it finer. The fineness of mince depends on the grinding process used during its production.
The texture of mince, also known as ground meat, is determined by the size of the holes in the grinding plate of the meat grinder. During the mincing process, meat is forced through these holes, resulting in the characteristic ground texture. Holding the mince under a pinch, or any other manual manipulation after grinding, won't significantly alter the size of the individual meat particles. If you require finer mince, you would need to use a meat grinder with a finer grinding plate or purchase mince that is already ground to a finer consistency. Some butchers offer different grades of mince, such as fine, medium, and coarse, based on the grinding plate used.
If you need a very fine texture for dishes like meatballs or pâté, consider using a food processor to further break down the mince after it has been ground. Be careful not to over-process it, as this can result in a paste-like consistency.