This answer is mainly focused on an older revision of the question. It quotes from and attempts to answer that.
I understand that this site has no community moderators
On the other hand, this is probably where the most engaged, smart, SE-savvy users hang out. We can meta-moderate pretty well. There's not that much difference between a 10k-er and a mod in terms of a lot of the powers we have, and a 10k-er actually has the advantage of not actually having to use a closehammer to close things or a one-shot delete.
If we did, I'm not entirely sure the actual problem you're facing might change, depending on their schedules.
Also, I ran into a situation where enough of my past flags, which were cast many weeks apart, were eventually handled and declined, and since the algorithm considers the handling time instead of the flagging time, I got banned from flagging because enough of my recently handled flags were declined. There was no advance warning of a ban because the flags were cast a long time ago and were handled at once.
It's worth remembering that a good way to deal with flags is, well, fire and forget. If a flag's not been handled yet, it's clearly not a priority. In addition, I do believe words were had before about flagging the same thing more than once. This might magnify the effects of your declined flags. Considering a past username of yours, this is kind of an antipattern that might have persisted over time, so... it might be worth reconsidering your flagging patterns.
If getting flag-banned is an issue, rather than blaming the time to handle, it might be worth relooking what you're flagging and why.
Or better yet, see if you can get sufficient reputation (cause I swear a lot of annoyances people have are with "shortcuts") and see if you can downvote and vote to delete instead.
I've thus resorted to pinging staff members in chat to get these time-sensitive issues handled quicker, but this has caused a lot of consternation among the community.
Few of these issues were truly time sensitive, IMO, and others and I attempted to feed this back many times. Folks were upset because the Tavern was one of the few places where we could hang out and talk with staff and your behaviour made that a lot more difficult. This was specifically pointed out, and ignored.
It's good to see you realise that folks were unhappy, and hope to see positive changes in your behaviour in the future.
There's other issues, but I feel that this isn't the place for them.
If something is truly time sensitive, and somehow there's no way to mitigate the issue with community meta-moderation, then pinging a CM might work. That said, even mods tend to ping specific users extremely rarely, preferring to find someone we know is on and free, or pretty much following the traditional flagging technique.
Pinging specific users makes demands of their time (which they may prefer to use in other ways) and makes chat a much less pleasant place.
requesting that a user who keeps reposting the same off-topic question be suspended
ehhh. Our current system, of requesting close votes and then delete votes on chat, is a suitable workaround. Almost tempted to see how much faster it is, but I personally feel Shog has many better things to do.