Thanks for the response! And it looks like you took care of marking these paths private. You have done the right thing, now the trails are tagged correctly, apps that use that data will render the tails showing their access.

You can read more here: https://support.alltrails.com/hc/en-us/articles/360058130612-What-if-a-trail-is-on-my-private-property

PS it might take a while before alltrails and other apps refresh their OSM data, so might looks wrong for a while in the apps before they update it.