Whether a tree is public or private is a matter of personal choice. Having had large sections of my tree lifted wholesale by non-relatives in the past , I prefer being private. It is, as you say searchable, so genuine researchers will find relatives in common.
I agree totally - FTM is designed to work with Ancestry. In fact it is their big selling point that you can sync your trees.
I like the versatility of being able to make notes and add images anywhere in the country and later when I get a signal, sync with Ancestry at my leisure.
Like Chris, I had vast swathes of my tree taken. I don't mind sharing, in fact I enjoy it. But when pictures I had taken and graphics I had made were on other's trees, without so much as a "by your leave", or "kiss my elbow" - I got angry.
So I created an impossible branch, where nothing and nobody could possibly have lived or occurred. Everything about them was wrong and so easily disproved. Didn't stop it being stolen though. After about a year of getting the Ancestry Hints about people added to my impossible branch in other trees, I made it private, but searchable. I have no regrets and feel no shame. If people are willing to steal your stuff, that's one thing. But not checking the data, then they get what they deserve.
Regards
Chas