This problem seems to be among the "eternal." As is known, Heinrich Hertz developed the formalism of mechanics without the time parameter. But if one reads his premises - how many assumptions on assumptions are there and how many of them are doubtful!
I checked the source for your quote of Einstein in the article: “People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion” and ended up here: https://www.christies.com/features/Einstein-letters-to-Michele-Besso-8422-1.aspx