I'm not sure what quality of integration Teams has - but Chat has had this integration for several months now.
Now, that document isn't just in that chat, its also in the OneDrive/Sharepoint and in the history of recently opened documents in something like Excel or Word or Powerpoint.
Like I said, it may have improved in the last year or so (about when I had to change), but last I used Workspace the chat in Meet was practically ephemeral. If someone shared a link to a document in Meet, at the end of the Meet that link was practically gone. No additional permissions was granted, I wouldn't be able to quickly find it in a recent documents list, no chat history in my recent chats, the next time for the meeting there was no chat history, etc.
The fact that Meet and Chat are still two separate "products" really points to that bifurcation of the two things. There's the video chat platform: Meet, and then there's the collaboration and chat tool: Chat. In the Google world if you want to talk by text you use one product, and if you want to talk by voice/video you use a different product, if you're wanting to schedule a meeting you use yet another product. But in the end a meeting is a meeting. Meetings can involve voice, video, text, documents, and more and usually benefit from scheduling. Needing multiple "products" means kludging things together or having things more un-linked. And in the Microsoft world there's a single tool that can do all of those things: Teams.
That for me is a huge drawback because it's constantly cluttering up a huge mess of different document versions in SharePoint and teams itself.
Try to search for the right one a few weeks later. Good luck :)
Don't have people copy and pasting the file all over the place and you won't have a thousand versions of important_numbers_v3_final_V2_DRAFT.xls.xlsx. Thats a people problem not a technical problem.
In the end we did need to use it because we moved from another storage service to OneDrive (another "quick win") for a manager and OneDrive is really bad at team-owned storage, it's really for personally owned files only.
The other side of Workspace is dealing with users who really want Office tooling for various reasons. Workspace and O365 still don't seem to play nice together. Workspace document management is great for Workplace-formatted documents, O365 is great for Office-formatted documents, and things often get messy when trying to combine the two.
I really liked Workspace and its really slick how integrated some of their document tools can be into Google Cloud so its easy to help team members who only really know spreadsheets get data in and out of cloud compute and UI builders and what not. But from last I saw of Hangouts they're still not as complete of a chat platform compared to Teams.