Is there any airport planning to install ILS CAT IIIc?
Landing is one thing, taxiing with no visibility whatsoever is another.
At the moment the answer is no. But new rules are in place that may one day allow it (or its equivalent). See this Rockwell Collins post: Worth the Wait — FAA’s New EFVS Rule FAR 91.176.

Main benefits to FAA’s new EFVS rule:

Enhances low-visibility flight and ground operations

(...)

Enables descent below Decision Altitude/Decision Height (DA/DH) or Minimum Descent Altitude (MDA) in low visibilities on a greater number of approach procedure types (precision, approach procedure with vertical guidance (APV) and non-precision)


It's still a new rule and thus the implementations are not clear yet.

RE: Why have specs and not use them?
The CAT IIIc definition may soon be removed:

[A] preview of proposed changes to the draft OpSpec and 14 CFR part 121 to date [was presented]. The definition of Cat IIIa, IIIb, and IIIc will be removed from the regulation. The FAA will continue to use the same definitions, however they will not be defined in 14 CFR. In the future, expect the NOS Charts to depict only one minima for Cat III.

It's unclear what that will entail. And what ICAO will do. So there's a chance the zero/zero definition will no longer be there.

Related: Is it true an ILS CAT IIIc can automatically taxi-in an aircraft?