Does Storage affect PC performance?

People say it's best to have at least 10% remaining of your storage for good performance.

Low storage won't slow down your system noticeably, (it may result in higher read/write times but you won't notice it.)
 
That can very much depend on what kind of disk you have. As spinning disks fill up they get slower as the reads & writes are hitting the inner part of the platter which is slower than the outer rings. There's also more seeking involved. It can also be fragmented.

If its an SSD with TLC NAND it could struggle some when really close to 100% full if you're doing large operations that require high I/O.