The answer to the title question is pretty straightforward. There's an obvious monetary return to increasing longevity of old, rich people. This does not require full success to be worthwhile - incremental improvements can be worth a whole lot.
In fact, one can easily argue that nobody really believes that aging research is the best way to reach 10,000-year mind-spans. Emulation/upload, or more likely non-biological minds to start with (all current humans die; future generations are all or partly electronic) seem like more likely paths to immortality for thinking beings.
Immortality for currently-living humans is pretty much _only_ likely through upload. Depending on timeframe, perhaps cryonics extends the time for any given individual before upload is working. Both early upload and cryonics+upload require some not-yet-identified economic situation to make it more desirable to upload/resurrect YOU vs cloning a much cheaper branch of a known-functioning em.
We've covered this topic quite a bit on the site from a lot of different angles. One of the most popular ones is "Whole Brain Emulation" or also sometimes called "Mind Uploading", which has a tag here: https://www.lesswrong.com/tag/mind-uploading
Even after solving aging, we're susceptible to death by accident, disease, and murder. Are there other research directions for achieving extremely long lifespans (at least 1,000-10,000 years)?
Even if they're not popular, does anyone have an idea they think would work? The best I've been able to come up with after thinking about it for 10 minutes is an isolated brain, which would require us to figure out how input/output channels to the brain work (or at least be able to reverse-engineer the ones we already have), keep an isolated brain alive and functioning reliably for indefinite amounts of time (we've made some progress here even with little interest: one, two), and focus all of medicine into curing diseases of the brain.
Anyway, please reply with your best approaches and ideas.