My cover pump will get clogged with seed pods occasionally but the little plastic cover on the bottom mostly prevents that with the leaves.
To get the cover off I get all the debris off as best i can by scooping & using the brush then add a weight near the pump so I get as much water off as possible. I try to get it cleaned when there will be several dry sunny days so the cover gets pretty dry & stays that way until removal. Then I use what I call the Taco

method (patent pending

) for my round pool. It requires at least 2 people but the more the merrier!
Keep the cover clipped to one half of the pool rails, fold the loose 1/2 over the clipped 1/2 creating a taco - any remaining debris/water is now your taco filling. The goal is to keep all the edges of the taco closed together until they are outside of the pool.
With someone at each side of the pool where the fold is hold the edges of the cover together un clipping the rest as you walk & pull the cover to the other side of the pool leaving the very center of the cover “taco”clipped for now. This will create a pocket of water etc- so put the pump in there until it’s as empty as possible. Then pull all the rest of the cover that is loose over the edge - with it pumped/scooped out there should be minimal debris left in the pocket. After all that is dumped out outside of the pool & the entire cover is on the outside, un clip the remaining clips. I very rarely get any debris or nasty water in the pool doing it this way. Before I had the electric pump I just used a little siphon one like this
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My kiddos have been the ones to help me do this for the past 8 years- their main job is to just hold the cover in various places & walk, not have brute strength. If it’s too heavy to pull then there’s too much water/debris & you risk a tear or dumping it all in the pool. I think that’s where people get impatient. Cleaning the cover & drying it out in advance go a long way in making it easier to deal with.
I often do a “soft opening” a week or so before where I have already refilled the pool & hooked everything back up before removing the cover that way I am ready to skim & all that right away once the cover comes off. I usually do the same for closing- just in reverse

Hopefully some of these tips make your opening/closing much simpler