Fungi are more related to animals than plants. Fungi produce "flesh."

Sea anemones are a group of predatory marine invertebrates constituting the order Actiniaria. Because of their colourful appearance, they are named after the Anemone, a terrestrial flowering plant. Sea anemones are classified in the phylum Cnidaria, class Anthozoa, subclass Hexacorallia. As cnidarians, sea anemones are related to corals, jellyfish, tube-dwelling anemones, and Hydra. Unlike jellyfish, sea anemones do not have a medusa stage in their life cycle.

They're a "plant" on the edge of creating a neural system while also producing "flesh."

Not really. These things have mouths and a digestive system. They are predators, have a nervous system, and they produce sperm.