What you are describing is essentially what the "First Post" review queue is there for. It's not a bot that reviews the posts, but users who have been part of the community for some time (500 reputation).
This review should of course be substantial, but the majority of regular reviewers use the "No action needed" very often. In many cases the question has serious problems, later gets heavily down-voted, and/ or closed. I am quite sure there is some explanation for this result; I don't know which. I don't think it's the average time spent with the review, as that does seem sufficient to me.
I asked much earlier that everyone is a bit more diligent with these reviews and more welcoming. (But this might be a discussion for another time.) From what I have seen in recent times, that plead went unanswered.
All is well to question the health of this system, but I really do not think that a bot could do the job any better. In fact, I believe the opposite will be achieved. I also think that most of the new users don't bother to create an account in the first place, a comment would simply go into the void, while the question meanders around until it eventually gets deleted. I suppose only very few read any of the links that might be posted in such a welcoming message.
Obviously, the best way to welcome and then retain new users is to post an answer; which is something a bot cannot do, and if it could, this website would be superfluous.
I personally think chemistry.se can be overwhelming the first time you use it. It is not the place where you can find a quick answer (to a new question) - something that most infrequent users seem to want. And I understand that the style we usually adopt here can seem rather unfriendly, as it is very brief and to the point. As unfortunate as it seems, this site is not meant to be an immediate help to the person asking. Instead it is a (rather well) curated database of interesting questions and answers. And that is exactly what it needs to be.
So while it should be a goal to get new people involved, and retain them, it cannot be a prioritised over the quality that needs to be maintained.
I do very much disagree that chemistry.se is a hostile place. I also don't think that a bot would do much change to that.
While I want this community to grow, thrive, and prosper, I don't think this can be done by an automated system, and if it could, I think it would be the wrong way.