Getting the easy part out of the way first, being the mod who migrated it to your site, I felt that it was on topic as it pertains to your FAQ.
Your point about the FAQ:
WordPress - Stack Exchange is for WordPress developers and administrators to ask questions about:
That pertains to whom the site is targeted for, but in reality anyone can and is encouraged to use the site.
This line is extremely subjective (as are all the site FAQs in regard to their target audience) and should not be used to determine whether or not it is suitable to migrate a question there based on whom it might appeal to.
Quoting your question:
Generally speaking, if the question can be asked and answered whether
or not WordPress is included (see: PHP, HTML/CSS, jQuery, server
configuration, etc.), then the question is clearly out of scope
As well as the title of the question you linked to itself (emphasis mine):
Cannot Upload Custom Header Image into Wordpress Dashboard
I don't see how that question can't be answered if Wordpress wasn't included; it involves Wordpress and administering it, the OP wants to customize the dashboard, which in itself, is used for administration.
Maybe the answer involves php and requires a programattic solution, but that doesn't mean that the solution is a general programming solution who's scope is more clearly defined on Stack Overflow. The context is overwhelmingly Wordpress in this case.
I appreciate if you don't agree, and if your community decides that it's not on topic, then please, close it as such (it's open as of this writing and has one off topic vote), and inform the moderators of your site to confer with us to remove the migration history.
Moving onto your more general points about building your site and site quality and migrations, that's more a discussion for your site's meta about the scope of your site.
Perhaps it could be more explicit, but the FAQ clearly implies that
user support questions are not in the scope of WPSE. Such questions
should be directed not to WPSE, but to the official wordpress.org
I don't believe this to be the case. By your own admission, it's an implication and I don't see that implied anywhere. There is also no mention anywhere about the Wordpress support forums in the FAQ.
When you ask what can be done to fix the problem, I'd recommend conferring with your colleagues on meta.WP.SE first to more explicitly define what is on and off topic for your site, as it appears that this isn't clearly defined in the first place and then, if necessary, codify it better in the FAQ.
The FAQ is the first line of defense for things such as this and it's what most moderators are going to go to when looking to see if something is on or off topic for a site. If it's not clear, then it's usually an indicator that the scope of your site is not clearly defined, and needs updating to reflect the general consensus of the community.
Granted, moderators who are doing the migrating can always ask the moderators of the target site, but given the volume of some sites, as well as the availability of other mods (or responses on the meta's target site), it's just not feasible.
This means that the target site really needs to recruit more users to do the appropriate cleanup work for those things that slip through the cracks. More high-rep users need to flag content for moderator attention, issue down/close/delete votes, etc.
Again, if that's something that's not happening on your site, then that's something that you should bring up on your sites meta, as a discussion on ways to handle the lack of participation of the community in common cleanup tasks that users engage in on the Stack Exchange network.
Here are some excellent examples of what was done by Zuly Gonzalez (a moderator) on the OnStartups meta site to encourage people to engage in performing more of the cleanup aspects of the site:
Now, you don't have to specifically offer monetary incentive, but meta posts like these help to raise awareness in the community to make the site better overall. I highly recommend reading them for tips on how to raise awareness to your own community about the issues it faces.