Why should they, when you can implement this yourself with a set of "shred boxes" each dated with either "today" or "today plus 30 days". Shred the oldest box today, make it today's new box, put papers to shred into that new box.

Repeat tomorrow.

Having the shredder automatically handle it in one compact unit would be more convenient and less error prone.

I think @al2o3cr has given the likely correct answer. Liability avoidance. If they "shred immediately" (as in pick up, transport, shred) their liability exposure is much less vs. offering a 'delayed shred' service.

Place the documents in a container and date it and then when the date passes to shred it then. No need for the shredder to keep track of this.

During that 30 days every one of those pieces of paper would be a liability for the shredding company - they'd need to be protected against tampering with the same degree of diligence as the client uses, despite 99.999% of them being destined for shredding.

There are also a lot more failure modes for delayed shredding. For instance, what if you asked for a document back and it couldn't be found? Could you be sure it was "shredded by accident" and not stolen? Will your regulators be happy with that answer?

In US healthcare (insurance), mailrooms may receive accountable mail i.e., mail pieces that can be tracked by sender (such as USPS registered mail, FedEx/UPS courier packages etc.,).

Mail is cut, opened, documents sorted, scanned and processed at the mailroom. But the paper is retained for X number of days at the mailroom facility and then Y number of days in an offsite storage facility and then disposed off through shredding. Mailroom staff move the paper that is ready for shredding into a shredding bin (that is locked; because HIPAA). Accountable mail is retained the longest (barring Legal which is even longer).

A shredding company brings an industrial heavy-duty shredder and picks up the shredding bins. A mailroom staff member unlocks the bin and supervises the onsite shredding.

That is the level of complexity that a simple deferred shredding requirement adds to a workflow.

Before you ask Why on earth are they keeping the paper? -- Remember that these are claims (and many times, claim appeals i.e., appeals against a previously denied claim; sometimes they come from lawyers on behalf of the insured); Imagine a mis-sorted document that causes an appeal to be delayed in processing. The insurance company is liable to be fined for each day of delay. Also, in case of accountable mail, the sending party has proof of delivery of the document, so if the case went to court, it is going to be a replay of the scenes from movie The Rainmaker and arguments on why the insurance company did not process the appeal decision on time.