We have a lot of larger pots and we don’t want to make them heavier, so we don’t want to cover the soil in the pots with gravel.
Anyway, you are right that some types of barriers will stop the airflow into the soil and that will harm the plant roots. Plants in containers need air in the soil. The container blocks the airflow from the sides, which wouldn’t happen to plants growing in the ground. If you use plastic wrap or aluminum foil on the top of the pot, you block the flow of air. A loose fitting of these materials is probably just fine. These solutions also can make it harder to water.
I suggest that if you are looking for a nicer-looking barrier that can block the cat but not air and water, you try window screen material. It is easy to cut with scissors and can be purchased in a long roll inexpensively. Depending on the plants in the pot, you can cut a slit or two to make it easy to install around the trunk. If you cut it in a square that is larger than the pot, the corners can fold over the pot to hold it in place. You could cut it into a circle that fits inside the top edge of the pot. If the cat pulls it off, you could try a crochet hoop to hold the screen in the circular shape and lock it into place on the plant trunk.
If the cat is really tough and strong, you could switch to using wire screen mesh—sometimes called hardware cloth. It has thicker wires and comes with holes one-eighth to one-quarter inch apart. It requires a strong cutter and leaves sharp points on the wires. Cut it larger than needed and bend the wires underneath to make the points safer.
