As we know, when an object perform pure rolling on a plane having
friction the friction acts on the edge in the direction opposite to
the direction of its translational motion. There are no other
horizontal forces except the applied force and friction.
Just to be clear, static friction only enables the wheel to turn without slipping when a torque is applied to the wheel axle.
But if, say, you take the wheel off a bicycle or a car and set it into rolling motion on a perfectly level surface, after you release it there is no torque applied to the wheel axle and therefore no static friction between the wheel and the road to oppose the force due to torque. The only friction acting on the wheel is rolling friction.
Now this friction opposes its translational motion, so the object
slowly stops rotating, but this friction provides a torque to the body
which should give it some angular acceleration
Ignoring air resistance and any friction at the axle, it would then only be rolling friction that slows the wheel and nothing to keep it moving, in the absence of an applied torque, except its inertia. Rolling friction is due to the inelastic deformation that the rubber of the tire experiences when it is in contact with the road. It in involves friction heating. The kinetic energy of the coasting wheel will be converted to rolling friction heating slowing the wheel down.See this article on rolling resistance from Wikipedia: https://en.wikipedia.org/wiki/Rolling_resistance
If this angular velocity is increasing its linear velocity should also
increase, but this doesn't happen. Why is this so? enter image
Your image is incorrect. The force marked "friction" on the image should be labeled "force caused by torque". A different line directed forward would be the static friction force that resists slipping between the wheel and surface allowing the wheel to accelerate. What you label "W" should be labeled "torque on wheel".
Bottom line: If there is no torque applied to the wheel (a coasting wheel), there is no static friction and the thing that keeps the wheel moving is its inertia. The thing that slows it down is rolling friction which dissipates heat at the expense of the kinetic energy (and therefore velocity) of the wheel.
Hope this helps.