A clogged downspout can be very expensive if it's not taken care of as soon as possible.
A clogged downspout can also cause water to run back under your roofline, causing water damage to the plywood under your roof and the soffit that holds the gutters by the roofline—all caused by the clogging of leaves from trees.
Pressurized water from a hose can flush the leaf blockage out easily. Plumbers' snakes have an auger tip that can cut through a blockage and lose some debris to start clearing the downspouts. The disadvantage is that you can also rip through a joint in the gutter system if you are not careful.
If the messy debris can be loosened up and then running up and down, the system can clear it so the water will flow freely. Electricians snakes work at the first elbow when the gutter curves under the fascia to push and or probe for any blockages, but not well for problems lower down in the system since all you do is push the debris down further into the downspouts.
Downspouts that go into underground drainage can be harder to ensure they are clear. If there is a cleanout access point at the ground or a way to take out the downspout, You can insert a snake or hose to check if it is clear.
Using pressurized water is still the safest, easiest, and quickest way to ensure you don't damage your water system. Once you get the downspout clean, be sure to let the water flow for a few minutes to get the water flow going. All the damage that sticks and leaves from trees can be avoided.
The water you can save can be used for many things such as watering plants, watering your lawn, and keeping them all growing healthy and fresh so they can thrive well.
Source of Information on How to Prevent Clogged Downspouts
