The Senate not only has the constitutional right to try Trump; it has a constitutional duty to do so.
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As
Article I, Section 3 notes: "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." If two-thirds of the Senate votes to convict Trump, only a simple majority vote would be needed to disqualify him from further federal office.
The prospect of disqualification is key here. If the Constitution's only punishment were removal from office, the GOP's constitutionality argument might have more merit. But as the text reads, senators would skirt their constitutional responsibilities by failing to evaluate
both possible consequences.
Ultimately, constitutional punishments, like criminal ones, serve both to punish and to deter misbehavior. The impeachment clause does not include an exception for an impeachment that occurs late in a president's term or that stretches into the next president's term. This would perversely give presidents incentives to behave especially badly right before they are about to leave office. Interpreting the Constitution's wording differently would serve only to shield potential wrongdoers from accountability.