I'm familiar with bodyweight exercises/calesthenics. I just don't feel my strength improving, though the flexibility is through the roof. I would like to be conventionally "strong" before I die, at least once!

If you don't feel your strength improving, you need to try harder exercises. Can you already do a one-legged squat? A one-armed push-up? A handstand pushup? A one armed pull-up?

Does it sound right that I think I'd need to lose more weight before I could do those? I'm 6'3" and "lumbering" is the right way to describe me. I only ever see small, light people doing those.

As with all bodyweight exercises, having less bodyweight makes them easier :). I'm 6'2" myself and always jealous of the 50kg asian dudes that can do all kinds of crazy things easily. However, I can do one-legged squats and one armed pushups and I'm working (for quite some time now...) on a press-to-handstand.