All you need to burnish is the shaving you have just generated.You have to remove the tool rest to make the operation safe,
get a handful of shavings and offer them to your work adding some pressure,you will generate heat .....so discard and get some more.
you can also use wet and dry ,say 320 grit followed by 600.
You will be surprised with the results.....good quality beeswax or carnauba wax will work beautifully applied with a coarse rag.It appears from you work that you are scraping which won’t give you the same quality finish as cutting,you will have to remove the fine rings before you burnish with 120 grit.
but remember,safety first,clear the rest and keep the rags small.