Does freeing up disk space improve the performance of MySQL?
Can a lack of free disk space impact MySQL performance?


Certainly a lack of disk space at all can prevent writes from happening. Especially when storage is beginning to become full (>90%, as a minimal ballpark) and the disks are fragmented, it can take some time to search where writes can even occur when MySQL is trying to write to disk. 

This would be true of any database platform or any application that writes to disk. This isn't so much a MySQL behavior as it is a storage behavior.

My suspicion is that you have different qualities of storage disks available to you filesystem(s) that contain the MySQL data. As your server became more full, you would be more frequently subject to the performance of your worst disks in terms of IO. 

After purging your server of old binlogs you've likely been able to increase your storage-tier's performance with IO because it takes a shorter amount of time to find places to write. Shorter time to find places to write means faster transaction performance. Faster transaction performance means less timeouts and noticeable delays in application functionality.