Version 1.0.0 started August 5th 2014. Which included the MLB players and the beginning as I see it. There was beta versions before which did not include MLB players or the initial version of one touch baseball. I started somewhere between the 2nd and 3rd week of August. So pretty close to the start.
I played for several weeks when the game didn't have any mlb players. Back then there was always quality free agents available to draft for a nominal amount of cash. I had an awesome team and when the update happened they replaced a lot of my great players with mediocre mlb players and I was quite ticked.
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Not sure when this game was launched. I believe I started playing in the later part of the summer '14.