iOS App Store keeps asking for password for free items
No. It’s not removed, but it can get stuck. I would disable this feature and then reboot if you think it’s already enabled. Also, free apps are basically the same as paid - you go through all the same hoops to “buy” save an optional second confirmation for payments. 

To allow downloading from the App Store without having to enter your password every time for free apps follow these steps:


Go to Settings
Scroll down and click Itunes and App Stores
Click Password Settings 
Toggle off "Require Password" under Free Downloads. 


You will now be able download apps without having to enter your password. 

UPDATE 

I since found the issue was localized to a bad home WiFi/router which blocked access to Apple servers. This can be verified by:


signing out of App Store in settings
trying to sign back in
getting an error message
disconnecting from WiFi, using cellular
signing back in
getting the “verify payment alert”
entering credit card Security code or selecting “none” payment type
this enables Touch ID for purchases