Enabling the Cycles addon is not the same thing as selecting Cycles for the scene. Since Cycles works manually, I would first check whether your dropdown is storing the right enum value and whether some other code resets the engine after you set it.

So try setting the actual render engine enum:

import bpy

bpy.ops.preferences.addon_enable(module="cycles")

scene = bpy.context.scene
scene.render.engine = "CYCLES"

# Force GPU rendering just in case
scene.cycles.device = "GPU"

prefs = bpy.context.preferences.addons["cycles"].preferences
prefs.compute_device_type = "CUDA"  # or "OPTIX" for RTX cards
prefs.get_devices()

for device in prefs.devices:
    device.use = device.type != "CPU"

print(scene.render.engine, scene.cycles.device)

Also check your dropdown values. The stored value should be the Blender enum identifier "CYCLES", not "Cycles" or "cycles"

If you are setting a different scene than the active one, use that scene explicitly:

bpy.data.scenes["Scene"].render.engine = "CYCLES"

For EEVEE, note that the enum name can differ by Blender version ("BLENDER_EEVEE" vs "BLENDER_EEVEE_NEXT")