OK, then it looks like total stack height from stove top to the top of the liner isn't all that tall. Getting the sweep to insulate the liner and isolate the stove heat from the masonry chimney will pay big dividends in more heat into the house, an easier-to-run stove when outside temps aren't real cold and draft is weaker, less chimney cleaning, and additional safety overall. It may seem like a lot of money now but this is a long-term investment, and the benefits of doing it right will be enjoyed for many years.
From the looks of the chimney pics, they may only have 12' of stack. Now, with 20' and an insulated liner and block-off plate, they probably
could run with the air closed, or almost closed.