You said you do not see a movement in MB51, what did you enter in MBST then?
How do you know that the stock was created with MB01 if you do not even see the movement in MB51?
As I can see that your batch has a SLED of 2010 it is probably even older in the creation date. There is no restriction in SAP to archive material documents. They do not depend on any other data which has to be archived first. So it is just the skills of the consultant which can make a system healthy in volume and performance or screw up system for any reporting and follow on activities like MBST.
MBST years after does not really make any sense, It is meant to be used close after a wrong posting. but if your material was long in the company and had probably undergone several physical inventories between 2010 and now, then you should use other movements to get rid of the quantity (or to make the batch zero)
Back to the archiving, Check in transaction SARA for object MM_MATBEL if there were some archive runs. Click the Admin button to see them.
If the system is healthy and well maintained, then you can go back to MB51 and activate the box for "short documents" , in this case SAP will read the archived documents from the index.
In case you just delete what you archive you will not have any chance to find the old documents.