A working inbox is a mixture of a few things that need action today, a larger number that need action eventually, and a great deal that needs none. The cost is not reading the important ones; it is finding them, repeatedly, among everything else. For businesses where the inbox is also the sales channel, that search happens many times a day and the important message is occasionally found late.
Tools that read and classify messages handle this well. Enquiries in one place, invoices in another, the rest below, with a short summary at the top of what arrived and what looks urgent. Because the classification is based on what the message says rather than who sent it, it copes with the case simple rules never did - a new customer, from an unknown address, describing a job, which is precisely the message a business most wants to catch.
The rule that keeps this safe is that triage sorts, it does not remove. Anything that files messages away unseen will eventually file the wrong one, and the failure is silent: a missed enquiry produces no error, no bounce and no evidence, just a customer who went elsewhere. Folders that a person still skims are recoverable. Automatic deletion, or a filter nobody ever looks at, is not - and the cost of the miss is much larger than the cost of glancing at a list.
It is worth being deliberate about what goes into these tools at all. An inbox contains customer details, financial information and things people said in confidence, and passing it through an external service is a decision with privacy and legal weight rather than a purely practical one. That is a reason to check where the processing happens and what is retained, not a reason to avoid the idea - but it should be a decision somebody made, rather than one that happened by default.
Used within those limits the gain is steady rather than dramatic. The urgent things are found sooner, the routine things stop competing for attention, and the inbox stops being a queue that has to be re-read all day. What it does not do is decide anything: the classification is a suggestion about where to look first, and the judgement about what actually matters stays with the person who knows the customers.