Thank you for your detailed and constructive feedback. While the machine will perform best in the dry, a higher height adjustment setting is needed in wet weather. The brushes should just 'tickle' the grass, as they are built to flick material into the hopper rather than scrape it in. This will reduce the force on the brushes, and the wheels shouldn't have a problem gripping the ground and driving the brushes.

I'm glad to hear that you're having success using the machine behind your finishing mower. For leaves, the sweeper works best collecting them when freshly fallen, and on reasonably dry, short grass. Based on your description of it "leaving neat lines in the grass", the brushes are likely set too low for when you're trying to collect leaves on wet grass. The brush height needs to be adjusted to 'tickle' the grass, and it's best to set them too high rather than too low. This lets the brushes flick the leaves into the hopper instead of the brushes manually scraping the leaves off the grass. If the brush is set too low, the force on the drive mechanism increases, which is likely why the wheels are cutting into the ground. Once the brush height is adjusted to the optimal position, the wheels can turn more freely, and you should have much more success!