the reality is more that it's handled by an intern using a fully GUI service to click a check box to unsubscribe you but finds even that a technical challenge.
The true reality from someone who works in this space is that every company with an intelligent marketing department uses an email service specialized to help market to a customer file filled with email addresses with preferences tailored to each customer, and marketing campaigns are designed days or weeks in advance and scheduled to go out on a rolling basis. Email addresses that get queued up for some sort of email campaign are already "locked and loaded" to some degree and are not designed to be unraveled to remove one random unsubscribe request. Since the email providers receive client updates perhaps once a day to fill campaign lists, your unsubscribe as a rule will not immediately sync to be removed from any future campaigns you're not already signed up to receive. On top of that, syncs can and do fail, however infrequently. This means that if there is a technical issue with the connection of some sort, it can take a day or two to resolve. A heavy marketer may still send a couple emails to a customer that unsubscribed while that sync issue is being repaired.
tl;dr: Most companies remove you from email campaigns within a day. 7-10 days is CYA language in case something goes wrong.
I also work in marketing. Im thinking about the people who instead of hitting "unsubscribe" or "manage preferences" will reply to the email saying "unsubscribe me" or use the contact us form to send us a message.
or use the customer support feedback option in the product to send a similar message
which, maybe we just don't pay for services that deal with that, but for us have to be manually entered
Depends how big your company is. Our product team wouldn't have the time to manage stuff like that, not to mention internal product isn't directly connected to the marketing tools without a reporting step