Code issues aside, every component in this set will be protected at, or below, its ampacity or rating. So if there are any real and present dangers in this installation, I do not know what they are.
On the other hand, if there are code issues to be resolved, I can say that the tap rules are not going to help your situation. The notion of the tap rules is that you can sometimes protect a conductor at its load end, rather than at the point at which it receives power. All such rules require protection to be installed at the load end. So if neither the floor 8 panel nor the floor 9 panel has a breaker, then their respective feeders do not meet the tap rule requirements.
My basic problem with this installation is in establishing the point at which a conductor receives its power. Consider the feeder to the floor 9 panel. It originates on floor 8, and there is no breaker on floor 8 to protect it. So the issue boils down to this: can we say that the feeder to the floor 9 panel gets its power from floor 7, and that the 250 amp main breaker on floor 7 is protecting it at its point of origin?
I think not. I think this violates the first paragraph of 240.21, and it does not meet the sub-paragraphs that follow.
One solution would be to install a 250 amp breaker within the floor 7 and floor 8 panels, to supply the feeders going up from those two points. Then you would not need main breakers at the upper panels.