Touch Screen Unresponsive Majority of Time

iPhone 6 Plus and to a lesser extent iPhone 6 are very prone to touch controller failure.

In other words, if it is out of warranty (otherwise take it to an Apple Store or Apple Certified Service Center), a new known good screen does not solve the issue, and there is no liquid damage in the phone, then the touch controller is probably failing. Could be some other part of the touch circuit, but intermittent usually means the touch IC (or controller) itself.

You'd want to keep backing up your phone as the touch function may stop completely at any time in that case.

Replacing the touch controller on the board is not a DYI project. The repair can easily kill the phone if not done properly using the right tools, skill, and experience.

If it turns out to be that, you can check my profile for a link to a website with reputed microsoldering experts who can repair it for you. This way you can find one near you, or take it to your local repair shop if they perform this kind of repair or subcontract it to someone else who does. Beware of newbies, it is a delicate procedure.