Why don't we use weights to store energy?

Another way to store energy in mass is the use of flywheels. You simply take a massive wheel and spin it up to store energy, use an electrical generator as a break to take energy out.

Their main limitation for use outside of the realm of professional settings where they can be properly monitored and maintained by experts is the potential to fail catastrophically.

Let's spin some numbers to further illustrate the poor energy density of gravity-based storage systems. Assume that you have a 100 kilogram lead weight that you can lower into a 10 meter deep hole in your yard.

Now, how much energy can it store? This is given by potential energy formula $E = mgh$, thus $E = 100\,\mathrm{kg} \cdot 9.8\,\mathrm{m}/\mathrm{s}^2 \cdot 10\,\mathrm{m} = 9.8\,\mathrm{kJ} \approx 2.7\,\mathrm{Wh}$.

For comparison, a single AA-sized battery stores about $2\,\mathrm{Wh}$ of energy.

We do. Just the weights we use are made not from lead, but from water. Many water reservoirs are also used to store energy by pumping water up when you have energy surplus, and letting it come down through the generators when you need energy. All you need for this purpose is two or more reservoirs at different altitudes.

We do, and we do it in the form of solid mass, too!

ARES uses rails to move concrete weights up and down.

Potential energy storage or gravity energy storage was under active development in 2013 in association with the California Independent System Operator. It examined the movement of earth-filled hopper rail cars driven by electric locomotives) from lower to higher elevations.

There is even an idea to use winches, as you described:

Stratosolar proposes to use winches supported by buoyant platforms at an altitude of 20 kilometers, to raise and lower solid masses.

Thus, your question is void, there is no reason we don't, because we do.

We do use mass in a gravitational field to store energy and have done for hundreds of years!

Grandfather Clocks

Grandfather Clocks have used powering weights since the 1660s. This was when they first gained their tall thin shape. The weights seen in the picture slowly descend as their stored energy is released. In order to add energy back into the system the weight needs to be wound back up.

Ffestiniog Pumped Power Scheme

The Ffestiniog Power Station in Wales was opened in 1963 and was the UK's first large scale pumped hydroelectric energy storage system. The reservoir works in a very similar fasion to other grid storage solutions. When excess energy is produced (because coal power stations can't be shut down immmediatly for example) water is pumped from the lower reservoir back up to the higher one. When there is a surge in demand water flows in the reverse direction back down through turbines smoothing out any peak demand spikes making the whole grid more efficient.

GravityLight

Finally something very similar to the design in your question. Here is a product which recently went through an Indiegogo funding campaign. It's called GravityLight and works by the action of a falling weight to provide power to remote villages and others who live off-grid.

Indeed pumped storage, ARES, and Gravity Light are all examples of using weights to store energy but, like an earlier post suggested, the mass / vertical height required (and resulting cost) makes many examples prohibitively expensive. With the right geography, pumped storage is a relatively inexpensive way of storing large amounts of energy, but it only really makes sense at large scales because of many factors, including the pipeline infrastructure and the distance from typically remote mountainous areas to centres of demand (affecting the build cost and cost of transmission).

To provide large amounts of electricity storage capacity for a long period (e.g. 20 years or more of daily operation), technologies like pumped storage and compressed air are currently much cheaper than batteries, though they are constrained by geography.

Other electricity storage technologies involving weights include those being developed by Gravitricity, Gravity Power (shown below), and Ground-Breaking Energy Storage (effectively cutting a large cylinder of earth and then raising it by pumping water underneath).

We can also use buoyancy as a means of storing energy. Research at the University of Windsor in Canada is ongoing into buoyancy energy storage, where a buoy is pulled down towards the seabed using a winch to charge the system, then allowed to rise back up to the surface of the sea in order to discharge the system and generate electricity. The winch is connected to a motor-generator in just the same way as the pump-turbine in a pumped storage plant is connected to a motor-generator, and can be located at the water surface (either on nearby land or on some kind of platform/barge) and connected to the base of the buoy via a pulley. Others are working on similar technology, including a company called Buoyant Energy.