How can I make a new drawing brush “fineliner” which can draw with a tip point starting and ending.
Grease Pencil.
There isn’t anything built in for 2.7x or the upcoming 2.8 (atm). If you really want something like this, you might consider either selecting these points manually and scaling down the thickness (alt-s), or writing a script that does this for all strokes.
OK I thought maybe it could be done with brush curves (Grease Pencil (T)