Einstein explained the dependence of stopping voltage on photon wavelength by quantizing energy of photon and in his model increasing the intensity will increase the number of photons keeping the frequency same and thereby not transferring enough energy to electrons to overcome threshold. Did he consider the possibility that electrons can keep on gaining energy from lower frequency photon and can eventually escape surface when they accumulate enough energy to overcome energy barrier. Fact that this is not observed experimentally may be due to the fact that electrons are not able to absorb arbitrary amount of energy and therefore electron energy levels are quantized and not the photon energy level.
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No, I don't think Einstein or his contemporaries ever considered the idea you proposed, because it would have been obvious to them that it wouldn't have worked. Einstein was proposing an interpretation of both the photoelectric effect and Planck's ad hoc calculation of the blackbody spectrum. Your idea would not work for explaining the blackbody spectrum. (Your idea also would not work for explaining the photoelectric effect, since energy is effectively continuous, not quantized, both for conduction electrons and for electrons in the vacuum. However, he wouldn't realistically have been able to figure this out in 1905.)