The Baltimore Sun, and The Atlanta Journal-Constitution do not follow that convention. For most of us this is about personal tastes. But some journals use this Gothic text to convey a notion of gravity, or deep seriousness. It's like they carry a banner implying "take me seriously, for God's sake! The NYT does have a mostly rich, influential urbanites that take themselves so seriously and expect the proles to do so as well.
What if the New York Times were to use a different type face. To take an extreme example, how about Comic Sans?
At least it's not boring, like Times New Roman.
Years ago a cartoonist, Walt Kelly of Pogo fame, chose to depict Deacon Mushrat* speaking in Gothic letters. Perhaps the Deacon aspired to an opinion column in the New York Times:
From Trumpies and ghosties
And long-leggedy models
And Politicians that go bump in the night,
Good Lord, deliver us!
*As cartoonist Walt Kelly styled him in Pogo.