Paper, such as towel paper, has a char point is somewhere between 424- 480F, depending upon the type of paper. Typically, it won't ignite until it gets to 480F.
100% cotton rags if untreated will char at a temperature similar to paper as both are essentially cellulose. If you use cloth treated with a flame retardant (such as that used for baby clothes), then the char point might be higher.
Another alternative is pure flax linen fabric that has a higher char point than cotton. Polyamide fabrics like Kevlar and Nomex have very high melting points. Nomex can tolerate up to 660F; that is why it is the fabric of choice for firefighter clothing. Kevlar is even more heat resistant.
While leather is often used for welder gloves and aprons, it cannot withstand temperatures any higher than cotton.
If your rags melted, it sounds like they were made of polyester rather than cotton. Many cleaning cloths these days are sold as microfiber which is cloth made of very fine strands of polyester. They are great for many uses, but conditioning a griddle surface is not one of them. Microfiber begins to melt at 140F; it cannot stand the heat.. Microfiber should not be used around a grill or griddle.
When seasoning a griddle, it is not necessary to have all the burners turned to high. You only need to heat the griddle to the smokepoint of the oil you are using. I use peanut oil which has a smoke point of 450F.The smoke point of Avocado oil is around 500F, which is above the ignition point of paper.
You do not need to apply oil to the griddle after it has already reached smoke point temperature. You can apply a thin layer of oil at a lower temperature and then raise the surface temperature until the oil starts to smoke. When the oil stops smoking, that layer has polymerized and it is time to allow the griddle to cook down to a more moderate temperature to apply another thin layer of oil.