The following theorem was proven by R.E. Tarjan in 1984:
Theorem (Sequential Access Theorem). If we access each of the nodes of an arbitrary initial tree once, in symmetric order, the total time spent is $O(n)$
But if we remove the smallest element one at a time, then does it imply that we sorted the element in $O(n)$ time? Of course not, because then it contradicts the following Theorem.
Any comparison sort algorithm requires $\Omega(n\lg n)$ comparisons in the worst case.
What was wrong with the way I tried to use the Sequential Access Theorem?