Let's not use the word 'end'; it has too many different meanings to you, and you seem disinclined to pick one and stick with it. Certainly, the number of years before any given moment are infinite; without number. They also (by your own definition) finish at a given moment.
There is no contradiction here; IF the past is infinite, then there are an infinite number of past moments - and yet there is still a choice of an infinite number of given moments which can arbitrarily be described as the end of the time that led up to that moment.
Your only objection to this seems to be that it doesn't feel 'right' to you - it makes you uncomfortable. But that's not reality's problem.
I have tried to explain it to you but you are
unable to learn thinking about it without prejudice, rather than enthusiastically joining me in my error. In terms of the number of years there is no difference in saying
infinite years
with no end or saying years with "no beginning". There in not one bit of difference in the number of years.
A count of the years in the past can begin anywhere.
But it has to begin somewhere.
So to count the number of years in the past you can start at any arbitrary moment and begin.
As I said, the first year prior to that arbitrary but necessary moment would be year # 1.
The year before that would be year # 2.
And so on...... forever. In a universe with "no beginning".
In a universe with no beginning the number of years prior to any given moment are
infinite without end.
To get to any moment in such a universe requires
an infinite number of years without end to have passed first.