It's hard to really pin down what your actual concern is from your question, but maybe I can provide some perspective that will be helpful. Please note that this is not an official Stack Exchange response. I'm just an (unelected) moderator of this site, and a SE user of some experience.
Executive Summary
It's essential for a site's success to have a broad user base. Critical mass is everything. A user should expect to be interested in some small subset of the total remit of a site. A site should be as wide as possible, but no wider. It's not necessary to make hard delineations between sites. It is not yet known how to correctly decompose knowledge between sites, but there are some hypotheses, and the experiments are underway.
Rationale
How do we know where to draw the boundaries of an SE site? That's a tricky question, with lots of grey areas. The Stack Exchange approach has been to "try it and see": proposals are made at Area 51, people try and refine scope, and if they reach critical mass, they launch. Importantly, these proposals, and the subsequent implemented sites, need not be distinct sets.
For example, a question about the programmer's text editor Vim might be on topic on Stack Overflow, Unix and Linux, and Ask Ubuntu. This is because knowledge in the real world has fuzzy boundaries, and doesn't fit into a strict hierarchy, as ontologists and the field of AI has come to learn the hard way, over a period of many years.
Board and Card Games itself has overlap. We allow questions about computer implementations of board games, even though these are also on topic at Arqade. We also overlap with the Poker, Chess and Go sites (the latter now defunct, and merged with B&CG). There is significant disagreement within the SE community about whether or not these distinct sites are necessary, what we gain or lose by merging, etc. etc. There are no easy answers.
So the real question is what does a site's scope gain us? The key here is community. You want a critical mass of quality contributors who can ask and answer questions well, so that the site can become self-sustaining, while generating high-quality content.
In your question you define three broad areas of game, and imply that there isn't much overlap between them. I disagree. People that like board games often play many different kinds of games. Take a look at the answers of our top contributors to see that demonstrated. The connecting thread here is the gaming - how to define interesting problems or representations of the world, and how to solve them well.
We make a distinction with video games, because the nature of the gaming, and the community, is completely different. We make a distinction with roleplaying, because while there is a fuzzy boundary, the big concerns of serious open-ended roleplaying games are significantly different (e.g. character development, immersiveness, creativity) from most board games.
Vocal communities for chess, poker, go, and even magic-the-gathering have all argued that they are 'special', that their game is particularly deep, that they are largely uninterested in other board games, and that experts will shun a site which does not exclusively cater to their game. Others (including myself) have argued that these sites are too narrow to succeed, that the Stack Overflow model proves that disjoint topics can co-exist, and that tags provide the customised user view. Stack Exchange is essentially doing the experiment.
The Magic proposal was shut, and all questions redirected here. That appears to have been very successful. The Go site was launched, but didn't have enough momentum. Its questions were folded in to this site, and rendering software for Go boards has been enabled for us. The Chess and Poker sites continue to exist in Beta for now, but have their own challenges, just as we do.
I think the key to a successful SE site scope is to be as wide as possible, but no wider. Stack Overflow has great scope because it allows any programmer to ask any technical question about programming, in any language. Arqade works because it covers all video games, on all platforms. I think Board and Card Games works because it covers all aspects of all board and card games.
It's really important to understand that most of the users of these sites don't interact with most of the questions! We all have our niches. On Stack Overflow, I only care about the two or three programming languages I use regularly. On Arqade, I completely ignore questions about Xbox games. I don't have the console, I can't answer them, and they aren't relevant to me.
Finally, I'll note that site scope is a work in progress. It's meta discussions here that help us to define what is and isn't acceptable, and that definition can and will evolve over time. For now, I see no argument for increased specificity. The site is working well, and the key for us to be launched is more users, not less.
