What do you mean by "standard 2-stroke oil"?
In my experience, most 2-stroke equipment will run fine on standard hardware store type oil- it may not be the best, but if the ratios are somewhere within the saw and or oil manufacturers spec, they might burn dirty, perhaps not to full performance and not be the most perfect mixed fuel to be running- but they should still run.
Now if you used a common "garden" variety oil in the mix at a ratio suggested for some kind of fancier full synthetic that is above the max recommended 50:1, you might have some problems.
Pull the muffler off and take some photos of the piston through the exhaust port- that will tell most of the story.
If it was indeed the fuel mix leaning the saw out, I dare say piston and ring/rings minimum and you might be able to clean the transfer of melted piston from the interior of the cylinder well enough to save it.
If there are gouges through or bits of plating flaking off the interior of the cylinder- you will need one of those as well.
About then it becomes a question of economics- is it worth trying to find and buy the repair parts, versus the cost of a new saw?