"All of these outer planets radiate away more energy than they receive from solar radiation, thus much of the winds must be driven by internal energy."
I submit the above reasoning doesn't hold water. To infer what the energy source is of the winds on the gas giants must come from other considerations.
In the case of the Earth the terrestrial flow patterns arise because the region of the equator has a larger influx of solar energy than the polar regions. The driver of the flow patterns is not so much the total energy influx, but the fact that the energy influx is uneven.
The difference in temperature between the equatorial regions and the polar regions drives formation of convection cells. The motion of the air mass in the convection cells is affected by the fact that the Earth is rotating. This leads, for example, to the formation of the prevalent winds that are commonly called 'the trade winds'.
In the case of the gas giants the migration of internally generated heat to the surface will not necessarily drive winds; the temperature will be fairly evenly distributed over the entire surface.
I did a quick search, and I did not immediataly see remarks about assessments of what the primary energy source is of the winds on the gas giants: temperature difference between equatorial regions and poles, or temperature differences between deeper layers and shallower layers.
I don't exclude the possibility that the energy source of the winds on the gas giants is the internally generated heat, but the reasoning you present does not establish that.
As we know, in the case of the Earth the angle between the plane of the Equator and the plane of the Earth's orbit is about 23 degrees. Hence the equatorial regions receive a much larger influx of solar energy than the polar regions.
As we know, this generalizes to all planets with the property that their equatorial plane is at an angle of at most tens of degrees with their orbital plane. (I get to Uranus a couple of paragraphs down.)
As stated earlier, the heat difference between equatorial region and polar region drives atmospheric convection cells. On a non-rotating celestial body that convection cell system would stretch from the equatorial region to the polar region, with warm air traveling in the upper half of the atmosphere towards polar region, and a return stream of colder air from polar region to equatorial region.
But the celestial body is rotating. When a system is rotating there is opposition to motion away and towards the center of rotation. When air mass moves closer to the center of (planetary) rotation the angular velocity of that air mass increases. This increased angular velocity prevents that air mass from continuing to move towards the center of (planetary) rotation. Instead of a single convection cell, stretching from equatorial region to polar region the atmosphere develops multiple convection cells, at various latitudes.
As we know, Uranus is an extreme exception to all of the above, with an angle close to 90 degrees between the plane of the equator and the orbital plane.
So: there is a point in the orbit of Uranus where one pole is pointing straight towards the Sun, and half a Uranus year later the other pole is pointing straight towards the Sun.
So Uranus receives energy influx from the Sun in a pattern totally different from the pattern of influx that all the other planets are experiencing. There is no comparison.
Returning now to the category of planets that have their axis of rotation close to perpendicular to their orbital plane:
If, with everything else the same, such a planet would be closer to the Sun, would atmospheric activity be more energetic?
I expect more energetic, but I's hard to guess by how much. The driver is difference in energy influx between equatorial region and polar region, and that difference increases less fast than the total energy influx as a function of distance to the Sun.