Ozone molecules have mass, and therefore exert gravity, but the atoms that make up the ozone layer would still be there and would still be exerting gravity even if they were something other than ozone. Doing something to the ozone layer would do nothing to its gravity.

Finally, the Earth's atmosphere is tiny compared to its rocky and metallic mass, so even if you stripped the atmosphere off, its gravity wouldn't change much.

No, not at all. Ozone or O₃ forms because of UV radiation from the Sun that zaps diatomic oxygen and makes a small amount of ozone. This has little to do with gravity!!! If the earth had much weaker gravity, then the atmosphere would not exist because the air would fly off into space. This is the case on Earth's Moon... the moon has small gravity and so it cannot retain an atmosphere.