Why does technetium not exist in nature?
Technetium does occur in nature. From the Los Alamos page on technetium:

Technetium was the first element to be produced artificially. Since its discovery, searches for the element in terrestrial material have been made. Finally in 1962, technetium-99 was isolated and identified in African pitchblende (a uranium rich ore) in extremely minute quantities as a spontaneous fission product of uranium-238 by B.T. Kenna and P.K. Kuroda.
Technetium has been found in the spectrum of S-, M-, and N-type stars, and its presence in stellar matter is leading to new theories of the production of heavy elements in the stars.

The journal article describing the 1962 discovery is Technetium in nature. B. T. Kenna and P. K. Kuroda.  Journal of Inorganic and Nuclear Chemistry 26(4) 493-499 (1964).