Here's the source.
https://ken-carylranch.org/wp-content/uploads/2021/05/KCR-Open-Space-Map-2020-11-web-version.pdf

As you can see, there are places where trail segments aren't directly labeled, so you have to guess the name based on nearby segments. I updated a few names after I was able to hike up to the area and see the way finding signs.

So that residents can see where they are while they're hiking. So that anyone looking at a saltine I map can interpret what they're looking at.

Whether or not you can think of a good reason, OSM is here to document what is actually there, not what you'd like people to see.

I'm sorry that Strava is not displaying data correctly, but that's something you should fix with them. It doesn't mean the trails names are secret.

I don't see how a trail marked private does anything more to encourage outside use than publishing an entire website dedicated to the open space.

Well documented-trails will encourage better users than having satellite imagery that shows obvious trails with no details.

Private land does not mean that trail names are secret. Please stop removing them.