Personally, I wouldn't worry about the small amount of chainsaw oil - your helpful composting microbes will likely decompose it into component parts.
However, fresh wood will be much slower to compost than most other things you add, as lignin is slow to break down. Leaving it out in the rain/snow all winter and then adding it next year, might actually improve its digestibility.
Personally, I have animals, so I let the ducks and geese poop on mine and then compost it. Also in the past, I've used it on paths around garden areas. It can enrich the soil in place, or once it has broken down for a couple of years, you can toss it up on the bed to add extra carbon. There's some modern evidence that we need more carbon in our soil in most places.
Lastly, I've also used shavings like that to make small batch biochar - which I generally add to my compost or duck runs. This is another great way to get more carbon in your soil, as well a providing local microbes with lots of hotel space. I do not chop down trees just to make biochar, but I definitely use waste-stream material to do so. We make it in a pot in our woodstove, so it heats the house a little.
https://permies.com/t/44894/Containers-making-biochar-wood-burning