No, there are certain types of questions that are generally considered off-topic across the entire Stack Exchange network because they're not a good fit to our Q&A system. In particular, questions that have multiple correct answers (polls), or those that invite mere opinions and extended discussion.
That's what the Gaming folks were really trying to tell you when they said that game recommendations are off-topic there. And not only are they off-topic there, they're off-topic everywhere. You can find a more thorough discussion of this topic in this blog article: Q&A is Hard, Let’s Go Shopping!
The reason I ask is because as the number of Stack Exchange sites increase, it seems questions like these could become more common.
Yes, that's true. It seems that there are a lot of people out there with these types of questions. But that doesn't make them on topic for SE. We've experimented with allowing these types of questions before, and it just didn't go well. Some people/sites still have battle scars from the experience.
Unfortunately, it isn't possible and doesn't make sense to just create a Stack Exchange site for questions that are off-topic on all other Stack Exchange sites. As you mention, there are other forum sites on the Internet that fill this void (pun intended). We simply can't be everything for all people. In order to be the best, you have to pick a goal or a focus, and then stick to it.
As far as the more general question here—i.e., where to ask a question that is on topic for a SE site, and is actually on topic for multiple SE sites...
You should take a look at and/or refer others to this recent blog article: Respect the community – your own, and others’.
Basically, if your question fits within the scope of allowed topics on one of the sites, then you should be able to ask it there, regardless of whether it might also fit on another site. Pick the site where you think you'll get the best answers (i.e., the one where you think the community includes more experts on your topic of interest), and then ask your question there.