I think what it comes down to is what you see as the "inbox" and what you see as the "archive".
With texts, functionally the Notification Center is the "inbox". Once you open the message notification (and presumably immediately respond), it disappears from the "inbox" and is stored in the conversation history. Functionally the conversation histories are the archive, so it doesn't really need to be archived again.
I think this is intentionally different from the way emails work. With emails, it's a very deliberate form of communication, and often long-form and complex, so you often can't respond to messages quickly. This means you need an inbox that you have complete control over so you can empty it at your own pace. But as a downside, this is also why email inboxes often get so backed up with messages. With texts, it's meant to be closer to a real-time conversation, with casual short-form messages, so the notification (without an email-like inbox) is meant to force you to respond immediately or as soon as you have a free moment.
I know no one likes to be forced, but if people are allowed to control texts like emails, then people will inevitably have backed up text message inboxes too. Also if the two forms of messaging function in the same way, what is the point in both existing? Only one should exist then. But if one is used differently and fills a different purpose, then it has a reason for existing.
Instead of archiving, I think what would be more useful is a quick easy way to forward a text message to your email inbox, for those text messages that require a more deliberate response at a later time.