Our Kenmore 70 series isn't always spinning. It will mostly spin a light load, as long as the tub is only half filled. The bottom of the agitator doesn't move and only if I press down and spin manually with the machine empty does it move at all. The machine also comes out of the spin cycle with a violent jerk that makes the whole Machine jump forward.
Most likely your issue is a WORN MOTOR COUPLER. The “sometimes” spin and load size you mentioned lead us to believe the coupler is worn out. If the coupler gets worn out, it will spin usually on small loads but not on large loads. Larger loads put more stress on the machine and if the coupler is shot then larger loads will not spin. Remove and replace coupler. Check YouTube for HOW TO video.