Top load Frigidaire built stack unit.
Just because you can turn the tub by hand doesn't mean it's OK, the lower one way bearing could be bad and locking up or the tub seal could be bad which would cause the upper spin bushing to gum up.
Do you see any signs of splatter inside the washer body at just the level of the bottom of the wash tub? If not then the seal and upper bushing is probably OK.
The belt is the clutch, (you have to use the correct OEM belt not just a regular auto/rubber belt of the same length). The belt is cloth covered and designed to slip until the tub is up to full speed. If the lower one way bearing is going bad and causing too much resistance to turning it will spin the motor pulley in one spot on the belt and burn the belt up.
You could also have an item of clothing that has got between the spin basket and outer tub that is causing it to not spin freely.
Turning by hand and the tub turns is not a good test unless you are experienced enough to know how one that is spinning correctly feels.
Your statement about it spinning OK if you put more tension on the tensioner leads me to believe that the transmission is binding up in some manner causing your problem. If the lower one-way bearing on the transmission is bad you have to replace the complete transmission, no parts are available to repair the transmission.
Source: WSM2700DAWWW will not spin