Spouting off to the NYT is not really private, is it?
It isn’t official, which makes it private.
Likewise, a priest (or a Pope) leading a 100,000 faithful in the Rosary at a stadium is still leading them in a private devotion. The same saying Mass “alone” is performing a public action, as he is exercising his priestly office.
One might question the prudence of spouting off to the Times, but better something stupid there than something stupid in an Encyclical.
If by not really private, you merely mean “potentially capable of causing grave scandal,” you may have a point, but the NYT is not where the plentitude of Papal authority is exercised.