you didn't say what kind of frame it is. Many frames have pressed in inserts and you might as well remove them if you can't live with the part inside the frame. If it's steel, it probably will work without the inside part.
I have cut down a seatpost on my lathe, I see no reason to have a seatpost that is that long down inside a frame just waiting to get stuck.
The item in second pic in your first post is called a rivnut and the best way to remove them is using a drill bit the same size as the hole in which the rivnut is secured. Drill deep enough that the head of the rivnut comes off and then push the remaing bit back into the hole. Turn your bike upside down to get it out of the seat tube.
FWIW , different in that its steel, Bike Friday adds the waterbottle brazeons
to the surface , behind the seat tube, uses a different piece,
the external is the hourglass piece, that often gets used for rack mounts,
brazed to the side of tubes,
but to the desired effect,
no interference in the seat post coming down.