Ok - let's examine some of the issues that require mods to contact SE right now, things which are inappropriate to bring up publicly on meta:
Destroying sensitive information
Frankly, I hate seeing much of this. It's a nice thing to do for people who inadvertently post their password or some such, but when folks are including contact information - email, phone #, etc - it's obviously intentional; edit it out, and leave it be. Unless the author wants it gone, who cares if it sits in the revision history somewhere?
Beyond that... Deleting and editing the post itself removes most of the time-sensitive nature of these.
Deep analysis or specialized handling for voting fraud
This badly needs to become more automated. Stuff the scripts don't catch is still pretty obvious in the mod tools most of the time. The current process wastes a lot of time, everyone's time; we're working on a better one.
It turns out this is a lot less common than we realized, and we'd like to make the scenario you describe even more uncommon: work on improving both the process and guidance for unregistered -> registered is underway, with various UI changes being tested seemingly at random on various sites as we speak.
That said, when it does occur, we'd prefer to see the user involved take an active role in fixing it; until they register, merging just kicks the can down the road. If it's causing problems and the person involved doesn't want to register, you can ask us to intervene, but again this isn't really time-sensitive.
I agree that chat is a lousy tool for these most of the time. It's pretty good for discussions, advice, and very time-sensitive things, but more and more of these requests end up being things that should just go into a queue.
Also, a lot of them are things that should be automated in some way once they reach a certain scale - vote-fraud comes to mind; merging too, to some extent. Chat has a nasty tendency to obscure the volume of such requests. Also, stuff gets ignored or lost...
I hate the flag idea mostly because I think of flags as being fairly light-weight. If there's a lot of context or research that's gone into a problem, distilling it down into a flag can be troublesome. They're good for the sorts of fairly obvious issues that crop up a lot on a site, things where the primary goal of communication is just to draw someone's attention to something... But if moderators on a site are unable to handle those sorts of issues, the site needs better tools, better education, or... better moderators.
My personal preference is to use email for complicated stuff that isn't particularly time-sensitive. You do get a queue of sorts, as well as the ability to convey a lot more information - in both directions.
Email has its own problems, but the simple solution here is to just add a "contact SE" link to every mod menu that collects information about whatever problem has cropped up and emails us. But that could just as easily feed into something on-site as well, which may give more flexibility for issues with different priorities.
As the number of sites and moderators continues to grow, we'll have to introduce something different to maintain effective communication. What that'll be depends largely on what we all need it to accomplish.