I guess you stumbled on a logical error (and you didn't read the article).
1. Very old light reaches us all the time, not just since the start of the Hubble. Thus light from very far away objects has hit Hubble from the beginning, but we weren't able yet to identify it. So there is a function of time, but it has more to do with our increasing ability to make sense of Hubble data.
2. The article talks mainly about the limits of Hubble. As it has a limited mirror area, the amount of light it can collect is limited. Objects farther away have to be brighter to be visible with Hubble.
3. Hubble works only with light that can be reflected by its mirror. The longest wavelength it can detect is 1 micron. As light that comes from far away is redshifted, its wavelength increases. Usually we use the Lyman series of absorbtion lines of Hydrogenium to measure the redshift. As soon as the shortest wavelength of the Lyman series is redshifted to a wavelength of more than 1 micron, we can't see it anymore in Hubble. Thus the farthest object of which we can estimate the distance with Hubble can't be farther away than the redshift of the Lyman series to 1 micron allows. Yes, also X ray can be redshifted to UV and to visible light, which then could be detected by Hubble, but we can't measure the redshift (yet), because we don't know how to identify the absorbtion lines that exists in X rays.