An impeachment trial is overseen by Congress and Senate, who both make the law and (in this case) sit in judgment on it, ignoring the modern legal principle of the separation of powers. Athenians would have been shocked, not because they believed in the separation of powers (their citizens too made the law in assembly and sat in judgment in the law courts) but because in democratic Athens it was citizens that decided verdicts, and in randomly selected and therefore unpredictable juries.
By contrast, two elected oligarchic cabals decide impeachment trials, usually making acquittal a foregone conclusion. Athenians would have been appalled: what was even remotely democratic about that?
Statistics make the point about Athenian citizen control over the performance of their leaders. The top ten political positions were the annually appointed military leaders, who also had other, e.g. legal, responsibilities. Of those top ten, over the 180 years of Athenian radical democracy, two a year on average were found guilty by the people on a capital charge. If they were foolish enough to be in Athens at the time, they were executed. Most were far-sighted enough to stay, and remain, well away.
One consequence of the politicisation of the courts was that Athenians were quite upfront about arguing the case from the political value of their benefits to society. As one said: ‘I have spent [on my financial obligations to the city] far more than was required so that you would hold me in high regard and, if I got into trouble, I might stand a better chance in court.’ Another defendant told jurors to remind their fellows how heroically he had served Athens in battle, unlike his opponent. Indeed, in certain cases litigants simply ignored the charge and argued about whether Athens’s best interests would (or would not) be served by an acquittal. Similarly, even as Donald Trump’s impeachment trial was progressing, he was boasting of his economic triumphs like any Athenian on a charge, as if it were an impertinence to impeach him, whatever he had done.
So the President seems unlikely to invite the Supreme Court to look into this oligarchic, undemocratic travesty of modern legal process.