What if kinetic energy of ejected electron = 0
Rather than debating whether an electron with no kinetic energy is ejected consider how an experiment to measure the photoelectron effect might be conducted. A photosensitive electron emitter is irradiated and the kinetic energy of ejected electrons is measured. One way to do this experiment is described in MIT Physics Department lab course notes. The setup is described in the following schematic:

A photodiode is irradiated with the filtered output of a mercury lamp. Only one of the available narrow bands of the lamp's spectrum is selected by the filter. The light strikes the K coated surface of the cathode of the photodiode. Emitted electrons are collected at a ring-shaped anode separated from the cathode by a gap, setting up a small current. An adjustable retarding voltage can be applied between the cathode and the anode to suppress the current, and the minimal cutoff voltage V which completely suppresses the current is measured. This voltage is theoretically related to the work function $\phi$ of the material and to the energy $h\nu$ of the incident light as
$$V = \frac{h}{e}\nu - \frac{1}{e} \phi$$
Now note perhaps the most important point with regard to your question: only a limited number of light frequencies are available in the experiment, corresponding to the bands generated by the mercury lamp. In addition the motion of a collection of electrons (a current) is measured, not the velocity of a single emitted electron. While it is possible to perform single particle experiments, at no point in the MIT experiment is the kinetic energy of individual electrons directly measured.
The lab notes also provide a short history of the discovery of the photoelectric effect:

Crude though the early data were, the qualitative fact of  the  dependence  of  the  critical  cutoff  voltage  on  the wavelength of light emerged with sufficient clarity to induce the young Albert Einstein, working as a patent examiner in the Swiss Patent Office in 1905, to link the effect with the recent idea, introduced by Planck in 1900,that matter radiates its energy in quanta of energy hν.

and

It was not until 1912 that the technical problems of making  precision  measurements  of  the  photoelectric  effect were overcome [...]

In other words, 7 years passed between the time that Einstein postulated his theoretical description and a confirmation based on &quot;clean&quot; data.
Note also from the lab notes how inherently &quot;messy&quot; such an experiment can be:

It is not possible to precisely determine the work function for removing an electron, because the cathode surface interacts with the remaining gases in the photocell as a getter, so that the surface characteristics change  a  little  from  the  ideal  case.   It  is  important  to note that the electronic work function φ is a material constant which incorporates the different emission potentials of the cathode and anode.  The former is a difficult quantity to estimate due to the manufacturing process which makes  the  cathode  surface  inhomogeneous.   It  is  composed  of  a  mixture  of  potassium,  potassium  oxide  and oxidized  silver.   For  this  reason,  you  need  to  take  care that the same area is always illuminated.  The emission work function of the photoelectrons can vary locally!