So the question is, "Can I use an electric winch with your anchor nest?"
The short answer is "Maybe".
The challenge with using an electric winch with your anchor nest, is that the winch will be mounted in line with the anchor nest and be pulling horizontally. If you have an anchor puller and buoy ball setup, it will be coming out of the water in a vertical position and will likely get caught up on the roller and not pull over your anchor nest.
To make this combination work effectively, you would need to have your electric winch mounted to allow it to retrieve the rode and anchor in almost a vertical position, or at least as close as possible to a vertical position.
This is why you don't see the combination of an anchor puller & buoy ball using an electric winch out on the water ways. Well, at least I haven’t seen any.
As you can see, it is not the anchor nest itself that keeps you from using an electric winch with your anchor nest, it's the anchor retrieval system on the rode that will keep you from using the electric winch with your anchor nest.
If you want to go the electric winch route, you may consider not attaching an anchor puller and buoy ball to your rode.
If you are anchoring in conditions that are likely to get your anchor stuck on the bottom, in rocks or other underwater obstructions, you will want to have a breakaway system on your anchor that allows you to pull up your anchor from the bottom. Even if you are using an anchor puller and buoy ball to retrieve your anchor, it is still highly recommended using a breakaway anchor system.
So, using an electric winch with your anchor nest, remains a big “Maybe”, depending on how you have your anchoring system set up.