The Constitution does not limit impeachment to incumbent officials.

That’s some rather impressive legal gymnastics, since the Constitution also says this:

The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.

-Article II, Section 4.

I’m fairly certain that a plain reading of that bit makes it clear that impeachment can only be invoked for sitting officers/judges. Especially since impeachment does not protect against criminal liability:

Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.

-Article I, Section 3

Beyond that, there’s the whole problem of them not being able to get 67 votes in the Senate (even assuming they could pass articles of impeachment in the House). All this would do is fire up Democratic voters, turn off independents and moderates, and likely demoralize Republican voters.