I would begin

Please help me to understand use of would begin and would have begin. Thanks in advance.

In my own work, my priority was also to try to get a sense of who each player
was. I would begin with an interview, in which I focused on understanding
a player’s background – when his talent was first recognized, how he
had been steered into pro football. In a second session,I would conduct a
more formal assessment to gain a deeper understanding of the player’s core personality, motivations, values, needs, problem-solving skills, and interests. Finally, in a third session, we would go through all the results of the assessment
tests. It was at that point we talked about who the player was, what really
challenged him, what put fire in his belly.

I would have begun

I would do something = a habitual action in the past. (before I retired, I would wake up, get dressed and leave for work at 6 am.)
I would have done something = a conditional action in the past. (I would have done something IF something else had happened)

would REPETITION IN PAST
When I was a kid, I would always go to the beach. When I was a kid, I wouldn’t go into the water by myself.
When he was young, he would always do his homework. When he got older, he would never do his homework

would
conditional 2. If I had been president, I would have cut the cost of education. [but I’m not president]
2. If I had been president, I would not have raised taxes.