There are three branches in government. Per the constitution they are separate with different powers. The legislature is to make laws, the judicial branch is to judge those laws and lawbreakers, the executive branch is to enforce those laws. Per constitution none if them are to possess the powers of the others. Therefore the president can not make laws neither can the judicial. The legislature can not enforce the laws or judge others. Therefore the chief justice of the supreme court sits in trial of impeachment. It is a trial in court and is required to be run as such
with the senate sitting as jury. In this case and only in this case as prosecutor they have the power of subpoena issued by the judge. In all other cases for them to subpoena they need to approach the courts. It is not listed as one of their powers. Read the constitution.