I have heard it stated that this was suggested when Impeachment was considered in the 1787 convention.
Would there have been more impeachments and some convictions if that were there?
You'd have practically needed a revolving door on the White House. Madison was quick to point out the objection. See
http://famous-trials.com/johnson/487-constitution
The Framers' Debates on the Impeachment Provisions (from the notes of James Madison, taken at the Constitutional Convention in Philadelphia, 1787):
It was moved & 2ded. to postpone the question of impeachments which was negatived. Mas. & S. Carolina only being ay.
On ye. Question, Shall the Executive be removeable on impeachments &c.?
Mas. no. Ct. ay. N. J. ay. Pa. ay. Del ay. Md. ay. Va. ay. N. C. ay. S. C. no. Geo. ay.
Saturday, September 8
The clause referring to the Senate, the trial of impeachments agst. the President, for Treason & bribery, was taken up.
Col. Mason. Why is the provision restrained to Treason & bribery only? Treason as defined in the Constitution will not reach many great and dangerous offences. Hastings is not guilty of Treason. Attempts to subvert the Constitution may not be Treason as above defined. As bills of attainder which have saved the British Constitution are forbidden, it is the more necessary to extend: the power of impeachments. He movd. to add after "bribery" "or maladministration." Mr. Gerry seconded him.
Mr. Madison So vague a term will be equivalent to a tenure during pleasure of the Senate.
Mr. Govr. Morris, it will not be put in force & can do no harm. An election of very four years will prevent maladministration.
Col. Mason withdrew "maladministration" & substitutes "other high crimes & misdemeanors agst. the State"
On the question thus altered
N. H. ay. Mas. ay. Ct. ay. N. J. no. Pa. no. Del. no. Md. ay. Va. ay. N. C. ay. S. C. ay. Geo. ay.