Add chiseled wood blocks: each wood type would resemble the head of a different mob, like the chiseled creeper faces of sandstone.

Each wood type would present as the head of a passive or neutral animal mob native to a biome where that mob is common. For example:

Birch: Pig

Oak: Sheep

Spruce: Wolf or Fox

Jungle: Parrot or Panda

Dark Oak: Cow

Acacia: Llama or Ocelot

If you want to include more mobs you could give stripped logs a different build path to chiseled heads. It would make sense for the stripped logs to lead to the heads of smaller mobs such as chickens, bees, and rabbits.

Or just make a menu of choices you could make while working with wood at a fletcher's carpentry table.

Chiseled ice blocks could represent dolphins, squids, axolotls, etc.

The stems of nether fungi could be chiseled into piglins (crimson) or zombie (warped) piglins.

EDIT: Vote for this on the Minecraft Feedback site.