"Adding more data is a good thing!"
Everyone here states this over and over again, but it is not universally true. By making these trails public you are essentially promoting them which changes the very nature of the trail itself. Mapping is not a neutral action. And this case, causes damage to the work of hundereds of people that has been put in over decades. 

"Access tags can be added to show that the trail isn’t public,"
Again, this is only useful if it is a) supplied at the start, which is very often isnt. and b) downstream data consumers show and/or respect the tags.  Which they dont. Users of other platforms only see the trail but none of the associated data. 
You might say, well thats on them to properly handle our data, but it is your naive approach to mapping these sensitive trails which is the root cause.  

"everybody benefit" - No, not everybody benefits for the above reasons. Please please think carefully about your actions as a community. 