That is utter BS.
He's right. A Senate trial is optional. It is NOT mandatory. The Constitution mandates:
1. The Senate is the only venue allowed to try impeachments
2. That Senators swear an oath for it
3. The Chief Justice of the Supreme Court presides if it is a Presidential impeachment
4. Conviction does not occur unless 2/3 of the Senators present vote for it.
It is up to the Senate to decide whether or not to have an inpeachment trial. They are free to decline to do so, and have actually done it once.
It is certainly within the power of the Senate not to hold a trial and to dismiss the charges. However, if it holds a trial, it has to hold a fair trial. An impeachment trial is clearly a criminal trial because the constitutional grounds are criminal (high crimes and misdemeanors) and therefore due process provisions of the constitution apply including the right to process for obtaining witnesses.