Is it valid to claim that one does not need to recite all the morning and evening remembrances (adhkar) found in "Hisn Al-Muslim" daily, and that it is sufficient to vary them, under the pretext that the Prophet (peace be upon him) did not recite all of them on a single day?

What you mentioned is incorrect. Rather, it is recommended for a Muslim to recite all the established ذكر (remembrance of Allah) if he is able. If he is only able to recite some of it, that is better than abandoning it entirely. The more important should be given precedence over the important, the better over the good, the sounder over the sound, and that which yields greater reward over others.