It bothers me that eating with hands or a spoon makes a mess at the table

Making a mess is an important part of building the ability to eat independently. Set up an "it's okay to spill" environment (like laying a sheet underneath) and let them use their hands and utensils to their heart's content. It's fine to introduce spoons and forks slowly, alongside eating with hands. This is a time to not rush, and to cherish the very motivation to eat.