I made a research and here's my guess:
1. The career always start on the day when your team must play the first match, but:
2. The game is unable to handle the situation when your team bypasses the qualification of the tournament (direct bye to group stage)
In case if your team goes directly to group stage, the game thinks that you're not in the tournament (at the moment when it decides the start date).
I forced TournamentCmn::AreEuroCupsAvailable to always return the true, and made some tests:
1. Arsenal : team plays in ECC quali round > game starts with ECC quali match
2. Barcelona : team plays in ECC group and Euro Superfinal, game starts with Euro Superfinal match, and you can play in ECC group because ECC starts after Euro Superfinal
3. Real Madrid : team plays in ECC group stage, game starts with La Liga match, because the game thinks you aren't in ECC.
I think that dumb logic of game start date is the reason why EA decided to skip euro cups in first season.