Disk errors in one of the SQL cluster nodes and SQL patching was failing with a lot of errors related to disk and permission issues
There is possibility of disk replacement, for detailed steps..  

Seems your team decision is correct as you have been tried to apply patches that been ran through failures.  

Edit (Including disk replacement steps summary from same source - referral link):


Take backups of all the databases including System DBs. [ONLY for disaster recovery]
Present all the drives from the new storage to the Windows Cluster. These disks must be visible under “Available Storage” Failover Cluster Manager
Now add these disks to the SQL Cluster Role/Group.
From the SQL Server dependencies, remove the drive letters pointing to the old storage
Take SQL Server Resource to an OFFLINE State
Make sure that the old disks should be online. Now start copying all the folders as is from old storage to the new storage. DO NOT change any files or folder structures.
Once all the files are copied, you need to swap the drive letters. Such that the new drives will get the old drive letters and vice-versa [This can be directly from Failover Cluster Manager] - This might not applicable in your case
Add the disks to the SQL Server dependencies
Make sure the SQL server service account has appropriate permissions on new disk/folder especially for the folders that are pointed in SQL Server - startup parameters
Bring the SQL Resource ONLINE.
Test connectivity from applications.
Once testing is completed thoroughly, disks from old storage can be removed from the Failover Cluster Manager