untermensche:
You are stuck in a loop caused by a misuse of language. The culprit, I think, is the verb "has passed."
To say that a century has passed is to say that 100 years have gone by since a point in time 100 years ago. Every time you say x amount of time has passed, you are implicitly identifying a span of time with a beginning x amount of time ago. There is simply no other way to parse its meaning. It makes no sense to say that 10 years have passed, without implying that the point of reference to that statement is 10 years ago. When you talk about the passage of time, it is always bounded by a beginning and an end. In other words, whether you say it or not, every utterance of "has passed" really means "has passed [since... ago]."
Therefore, it's not an error of logic to say that an infinite amount of time has passed, it's an error of language. It's literally a meaningless statement; the reference point is undefined, so there is no way to interpret it. There is no coherent concept to which the proposition refers. Just as a square cannot be said to have circumference, an infinite amount of time cannot be said to pass or not.
So, the argument that time cannot extend infinitely into the past because if it did, an infinite amount of time has passed before today, contains a string of nonsense masquerading as a meaningful phrase. It would be the same as saying time cannot extend infinitely into the past because if it did, scarecrow funk explosion without falafel administrator.