Why does it take two weeks to process an unsubscribe?

Why does it take “10 business days” to process an unsubscription request?
It almost never does. An unsubscription request will often take effect instantly and it would be rare that it would take more than a few business days.
So why do some businesses say your email address will be removed “within 10 business days” when they know it’ll be almost immediate?
It’s better to underpromise and overdeliver. No recipient is going to be annoyed if they stop getting mail sooner than they were promised. But tell them they’ve been unsubscribed and will receive no more email, then have mail from you end up in their inbox the following morning and they may get mad.
Why “10 business days” in particular?
The US CAN-SPAM act says you have to honor unsubscription requests by then, so that’s the upper limit to what’s legal:

…
If a recipient makes a request using a mechanism provided pursuant to paragraph (3) not to receive some or any commercial electronic mail messages from such sender, then it is unlawful –
(i) for the sender to initiate the transmission to the recipient, more than 10 business days after the receipt of such request, of a commercial electronic mail message that falls within the scope of the request
…

Why would it ever be anything other than instant?
Sending bulk mail isn’t usually just a case of taking a list of email address and a message and hitting a button. More often you’ll do something like export a subset of email addresses from your database, depending on who you’re targeting your message at, then have someone double check it, then upload those addresses to the ESP you’re using to send your mail in time for a regularly scheduled weekly mailing. Then, when someone hits the unsubscribe link mail to them may be suppressed by the ESP immediately, but those unsubscription requests will also need to be passed back to the list owner so they can suppress mail to them sent from elsewhere.
If you’re regularly sending out a mailing on Wednesday morning, you’ll want to have your recipient list ready to go by Tuesday afternoon at the latest. More likely you’d want to give yourself a days headroom to fix any problems – “stuff just goes wrong sometimes” – so you might aim for Monday afternoon. You’ll need to extract the email addresses and some metadata about the customer in batch, have someone in another group sanity check things then, if needed, extract the addresses and metadata again. In a large ecommerce system that’s redlining just handling peak transaction rates you may well push that to off hours, which would mean you’d extract the addresses Thursday night, have them sanity checked Friday and redo the extraction Friday night, if needed. Doing anything Friday night is doomed to failure, so aiming to do the initial extraction Wednesday night is more plausible. You’d download the suppression list from your ESP just before doing that initial extraction, on Wednesday morning. That could easily lead to a five business day latency between an unsubscription request and mail still being sent to the recipient. And that’s in the ideal case, where nobody is taking vacation, everybody does everything right and so on.
So some businesses may well be able to suspend mail to a recipient immediately they receive an unsubscription request but more complex ones – those that use multiple ESPs, segment their customer database for targeted mail rather than just doing a “batch and blast” to everyone, and have humans in the loop when deciding who to send email to – may take a little longer.