Of course taking a 2010 computer to an AASP will just result in them saying they won't even look at anything that old since Apple won't look at anything older than 5 years unless required to do so by law (e.g., California).
Vintage and obsolete products - https://support.apple.com/HT201624
"Vintage and obsolete products
Owners of iPad, iPhone, iPod or Mac products may obtain service and parts from Apple or Apple service providers for 5 years after the product is no longer manufactured (or longer where required by law). Apple has discontinued support for certain technologically obsolete and vintage products.
bigel82: It isn't at all unusual for optical drives to fail after 5 years or so. For one, they are idea places for dust to collect. You can try a drive cleaning disc but I have rarely had those resurrect a failing drive for me. I simply buy an external optical drive for maybe $20-$40. The main issue with those is if you use Apple's DVD player, but there are workarounds such as using VLC media player instead.