Keep in mind that there is a difference between
parliamentary government and ours. The president
is both leader of government and head of state.
The prime minister is merely the member of
parliament elevated by his party to run government.
Since s/he does not represent that state (that is
the Queen's function) it is much easier to get
rid of a PM, usually by a vote of no confidence.
The tsunami of removing an impeached president
from office, as we have seen, is very difficult.