It's not just liability, but also the "cost of extraction".
There is no such thing as "your on your own", someone will eventually call 911 and then the authorities are mandated to respond. They have to rally all the EMS, and now since it is the middle of the woods it becomes a SAR mission. That becomes expensive, and it is the town, not the park that provides emergency services. Given the budget constraints the towns and parks are under, you can expect such a request to be easily dismissed.
As mentioned, some chapter/park leads negotiate scheduled rides, and so they have time to plan ahead, ensure they have sweeps, take headcount and make sure they have a time buffer to get everyone out.