I'd not like to see this option getting implemented.
◉ It is not practical. Feeds of any kind will consume transcript space, and therefore randomly interrupt conversations, if any. This will not encourage conversations, nor provide more opportunities for users to engage with each other. For a more practical example, come take a tour with me into Java Sucks!
This is a quiet room. Users don't chat often. Therefore, the Newest questions tagged singleton java - Stack Overflow feeds exists, and dumps quite frequently:
>>> Posting questions (and answers) of the users in low traffic rooms will not encourage conversation, and therefore will have no effect unless you want a heavy transcript.
Now, to take another case, not every room is quiet, and some rooms are populated and active. "Chat regulars" are in place, but few would bother participating on the main site - Which is why they are "chat regulars" on the first place. Posting questions (and answers) feed items will not get any attention - They simply float past, no one looks at them.
>>> Posting questions (and answers) of the users in high traffic rooms will not motivate any of the chat users to visit the main site because of this.
◉ It creates a motive to substitute normal on-site participation with it, and therefore creates a "friend ring". We've went on this many times before; Previously, previously, previously, previously.
◉ Not everyone in the room is interested in every single post you wrote. Sites covers many fields, as indicated by the wide varieties of tags. Most of the feeds posted could be irrelevant to one's interest, and therefore doesn't encourage participation effectively, no matter how it's done.
◉ Stack Exchange is a big city, and you cannot assume everyone wants to get to know each other until they show an interest to. Automating this is too far to go.
Against the points:
Chat is often disconnected from what's going on on the main site
◉ As seen from Privileges - talk in chat, "All sites have a real time web chat component, or as we like to call it, the 'third place'." The third place is never meant to be related to the main place, unless you like your boss to discuss your work with you on every lunch session. I doubt it.
while we're chatting, we're not creating/improving content
◉ Having feeds does not encourage creating content. Arguments can go on this point, however having feeds posted which gives another bonus slight bit of attention simply isn't any motivation. You can host a tiny contest on the most upvotes earned monthly, or take turns writing posts, but feeds doesn't do any of this for you. It is perhaps more constructive for one to implement a mailing list which sends you the posts submitted by all the users in this room.
◉ To put it in a clear way, this doesn't encourage chat users to participate on the main site and therefore to create or to improve content.
◉ This only adds the focus onto the users, and therefore causes the feeds to be biased towards the same users. As a result, the feeds shows posts only by the few users - This will at most dump a rise of attention on posts made by the chat users in the room, but how does this contribute to motivation to create new content at all? It doesn't.
One practical result of having this feed will be that users active in chat who post will get more votes.
This, my audience, is why this feature should never air - it introduces voting rings.
I don't know if that's fair or not, but I'm pretty sure it'll result in an increase in constructive engagement on main.
◉ Vote more
!= constructive engagement. Vote more
!= encourages further engagement. Vote more
!= directly causes further engagement.
Of course the feed might encourage more users into chat, but I don't think that'll be counter-productive as it'll only be users arriving in chat because they're creating content on main.
Your premise is invalid. If users arrives in chat solely because they're creating content on main, then this shows help vampires are lurking to attract attention onto their questions (which were of low attention for reasons including but not limited to poorly written ones, closed ones, obscure ones that no one figured out what the self-invented terms meant, et cetera...), not because it encourages chat participation.
Therefore, it is counter-productive.
If it's abused there are already all the necessary tools in place to combat that abuse: kicking users, deleting messages, turning off the feed etc. Having a higher rep limit for the feed is going to make this less and less likely anyway.
◉ Only room owners and moderators can perform the actions you mentioned. Therefore, this feature introduces a need for moderation, which increases channel pressure as users are being punished for abusing.
◉ Reputation is not a measurement of whether users can use a feature well. Nor experience. Of no degree of reputation limit to kick in the feature will it deem a lack of abuse.
- Userscripts. If you really care about a certain user, build it yourself. Or get someone to build it.
- Share it (the posts) by yourself(ves). Just make it clear that sharing what you write is encouraged and isn't blatant promotion. This works, and most rooms stick to it, even the ones I own.