Probably true for the US. In Europe, it was very country-dependent. Here PS2 sold simply because it was the new PlayStation, to a public that for the vast majority wasn't even aware there was competition.

Weren't a good chunk of DVD players almost as expensive as the PS2? You basically got a free gaming console with your DVD player.

This. Thrifty parents with no interest in gaming saw the PS2 and thought "Sony DVD player". The GameCube was merely an expensive Nintendo time sink. To this end my brother and I took out a loan to buy out GameCube but could have gotten the PS2 for free for Christmas - we wanted to play Super Smash Brothers et al that much and knew that we couldn't avail ourselves of the PS2's better-selling titles anyhow due to their M ratings.

As I write this it does feel like both Sony and Microsoft really started to push the whole living room entertainment convergence thing around this time while Nintendo happily stayed in their lane. The same dynamic continues to this day.

I bought the PS2 because it had a DVD player absolutely. Same thing for PS3; my only Blu-Ray player.