I get the channel here. It's called Kids Thirteen in my area, and has existed since 2002. Anyway, it seems to be the same schedule as it was before the relaunch, which has a heavy focus on Thomas and Friends and Sesame Street. I actually remember watching Kids 13 pre-relaunch, and usually they would air preschool shows in the morning, stuff aimed at older kids in the afternoon, and shows for parents like A Place Of Our Own (which I watched on the main Thirteen channel as it came on before PBS Kids) at night. In between shows would be specially made promos for the shows on the station. For example, there was a Sesame one that started with Grover saying "Aloha!" and a kid's voice asking "Are you ready to go somewhere exciting?" or something, a Barney promo which had a scene where he stood next to a treehouse and a Mr. Rogers promo that began with him looking at the trolley. I could only find this Kids 13-exclusive promo online:
And this isn't the first PBS Kids cable channel. There was an attempt to bring a PBS Kids channel to cable in the late 90's/early 2000's, but it failed because it was only on one cable provider (DirecTV) and some shows, like Sesame Street, were not on it due to other cable stations having the rights to them (Noggin had the rights to Sesame at the time).
Here's an article about it.
I don't see why they don't show Mr. Rogers Neighborhood. We need his lessons now more then ever.
I did see one local PBS that had a kids' station that showed it around the time this channel premiered...at 5:30AM (at the time, episode 1520 was on). This was followed by an airing of a Season 1 episode of Arthur (the one with "Meek for a Week"). I forgot which channel it was, though...