If a newspaper or any other media source takes a side on impeachment, it cannot be objective when reporting on impeachment.
True.
Around the time Elvis was recoding Heartbreak Hotel the news media used to at least claim they were reporting what they found about all candidates even while unconsciously shaping the stories with omissions. They used to say "But according to a spokesman for his opponent the explanation is not that simple." They don't usually pretend any more, with a few exceptions.
Now they start with "As with other debunked and disproven conspiracy theories, Trump's apologists today made negative references to Hunter Biden and Adam Schiff that do not stand up to fact checking."