  In this paper, the third and final of a series, we present complete K-band
imaging and some complementary I-band imaging of the filtered 6C* sample. We
find no systematic differences between the K-z relation of 6C* radio galaxies
and those from complete samples, so the near-infrared properties of luminous
radio galaxies are not obviously biased by the additional 6C* radio selection
criteria (steep spectral index and small angular size). The 6C* K-z data
significantly improve delineation of the K-z relation for radio galaxies at
high-redshift (z >2). Accounting for non-stellar contamination, and for
correlations between radio luminosity and stellar mass, we find little support
for previous claims that the underlying scatter in the stellar luminosity of
radio galaxies increases significantly at z >2. In a particular spatially-flat
universe with a cosmological constant, the most luminous radio sources appear
to be associated with galaxies with a luminosity distribution with a high mean
(~5 Lstar), and a low dispersion (sigma ~ 0.5 mag) which formed their stars at
epochs corresponding to z >~2.5. This result is in line with recent sub-mm
studies of high-redshift radio galaxies and the inferred ages of extremely red
objects from faint radio samples.
