Can you post any evidence for this? I somehow have it in my mind that it's the other way around (i.e., mean doesn't grow anymore, but 99th percentile still does).

You can't make any conclusions about LEV by just looking at current stats.

Medicine in its current form is just not prepared to deal with these things. It evolved to deal very well with infections and injuries, in both cases helping the body to heal itself, which it wants to do anyway.

Aging is different. An aging body actually wants to destroy itself. It intentionally disables self-repair mechanisms, allowing the diseases of old age to take over. It's the first time we have to override these internal goals. Treating these diseases individually, like we do now because it's the best we have to offer, is a losing game.

escape velocity in this context means that you stay at least one step ahead of dying, through medical advances, to the point that the first immortal human (at least one who will not die of old age) may have already been born.

Not the OP but I believe they are referring to the idea that if longevity research causes lifetime extension to be possible, then some folks think they can then avoid age-based death entirely. I think that the crux of the argument is that once we find "THE reason" why we age and if we can halt/end that process then we have beaten senescence.

I personally think this view is too simplistic about how our bodies work and overly optimistic that we a) won't hit other limiting factors that aren't hit today by ~100 and b) that interventions to stop senescence won't have unexpected side effects.

metabolically, yes