Actually I don't think that a Sent box would help. If these messages were in your SMS database they'd have their own conversation, even if all of the messages in that conversation were outgoing ones (just like if you send messages to a new recipient and they haven't replied you can still see a one-sided conversation with them in your message list). So if they don't appear in your message list at all, reorganising that message list into "received" and "sent" wouldn't make any difference.
But it is very surprising that ATT can't identify who the messages are being sent to. I suspect I know what the answer would be, but can they block the messages? Otherwise it is a case of identify apps that are capable of sending and try to rule them out one-by-one. Do these messages go at a particular time of day? That might make it easier to do.
On the question of keeping business and personal message exchanges separate, I think the fundamental thing to understand is that SMS systems don't include "threading". There's no difference between replying to a message and sending a new message to that number, there is no record that says "this message was in reply to that one" or "this is the start of a new exchange", it's always just "phone X sends a message to phone Y". When your SMS app shows a "conversation" it may look like a series of replies to earlier messages, but all it really is is the set of all messages involving just those 2 numbers, ordered by time. This is why you can't "start a new conversation" for a business discussion, because that's just not how SMS work.
(I'm also not sure the old system really is better for this. Your problem is that the "conversation" mixes business and personal, but then so does your "received" box or "sent" box. It's just that the old way also mixed them with messages from other people as well. So either way you have to read through to find the exchange you want, but in the old way you can't see both halves of the exchange at the same time, and you may have other people's messages popping up inbetween the ones you are looking for. I know why you want it, but from a practical point of view I think it's easier to find and review a particular discussion in the "conversation" paradigm than the "sent/received" one).
But can you separate them? If the person uses different phones for business and personal then it's easy: rather than having both phone numbers in a single contact, create a separate contact, e.g. "John Doe (business name)" for their business number, and then the messages will end up in different conversations. If they use the same number for both then I don't see how any app can help, because for the reason described above the idea of starting a new conversation with the same number is just not how SMS works. So if you really want to keep business and personal separate, and they are the same number, all I can think of is using different messaging systems, e.g. WhatsApp for business and SMS/MMS for personal: those are different types of message, so Google Messages cannot receive WhatsApp (or Signal, Telegraph, WeChat or whatever), and as long as you don't let the other app send/receive SMS as well (because some of them can) they will be in entirely separate apps.
The one I'm not sure about is this "no way to reply without the attachment", because there I'm not sure what you are describing. For Google Messages "attachments" presumably means MMS (or maybe RCS "Chat") messages, as opposed to SMS. I've not used MMS for almost a decade: it's much more limited in what it can send than any other messaging system, but in the UK still remains expensive (while SMS have been effectively free for a long time), and so I know nobody who still uses it. But historically I have replied to MMS containing media without my reply including a copy of said media, so have to assume that you mean something else when you say this. Of course I've never used RCS (I know nobody who has), and don't use Google's Messages app, but for MMS I'd expect Messages to work the same as any other app. Hence I think you must be talking about something different from my experience.
