If it was getting draft, it should suck the smoke up.
That's not how it works.
As hot gasses lighter than outside air rise in the chimney, it creates a low pressure area in chimney, pipe and stove. The lowest pressure is measured at the stove collar, called draft. The greater the temperature difference between inside and outside of chimney, the greater the pressure differential. (less differential than in winter) This lower pressure in the stove allows atmospheric air pressure to PUSH into the stove. This is what feeds the fire oxygen.
Since the chimney creates draft (low pressure), everything else takes away from it, adding resistance to flow, reducing draft. The cap or screen at top, reducers, straight pipe, elbows, tees, and a flue pipe damper which is a variable resistance. The most resistance is the air inlet setting, and a baffle in the firebox ads internal resistance.
The larger the chimney flue diameter, the more
capacity the chimney has, meaning it can make a larger stove work. Since the chimney is the engine that drives the stove, you have a small engine, probably clogged even smaller than 6 inches in diameter. The square inch area of your chimney flue is 28.26 inches. (if round) Depending on size of stove, if it requires 8, that is 50.24 square inches. 8 to 6 may not seem that much, but it is half the size you need for maximum heat output.
Square inch area is calculated by radius squared, (1/2 the diameter X itself) x pi or 3.14. So 6 inch is 3x3=9 x 3.14= 28.26 and 8 inch is 4x4=16 x 3.14= 50.24. This is the calculation needed for baffle size.
The space smoke travels over and around the baffle is critical. It can not be smaller than the square inch area of stove outlet, pipe and chimney flue which should be the same size. In your case no smaller than the chimney flue square inch area. This will give the stove reduced capacity since it was designed for a larger chimney. So measure the "smoke space" or area the exhaust travels around baffle. Length x width of space giving you the square inch area the stove has to exhaust. Originally it would not be smaller than 50.24 square inches for a 8 inch outlet. You are already choking the stove down with a smaller flue, so you can't be smaller than flue size.
The velocity coming in is what mixes the flammable gasses coming out of the wood mix with oxygen. When kindling is lit, it should roar up the chimney, quickly igniting larger wood as it is added.
You may be able to clean your chimney from the bottom with a chimney whip. Look into a "Soot Eater" which is flexible rods, with what looks like a string trimmer head on the top. It takes a drill to rotate, so if you don't have electric at this area, you would need a battery operated drill. You will be amazed how much better this does than a brush.
Inspect the chimney flue at least yearly. Burning below 250* to the top allows water vapor from combustion to condense on chimney flue walls, allowing smoke particles to stick. This is creosote which can accumulate rapidly when other factors allow the flue to drop below 250* while smoke is present.