It's not the duty of that "new feature" to care about what the manager does with it. It's about giving us the tools, the responsibilty what you do with it is still up to the manager.
I don't expect the new feature to care, however the developers need to. I think encouraging financial irresponsibility would be a bad idea. Why do new users get extra money for several weeks, instead of one big lump sum. Because they'd spend it badly.
In your idea, you're talking about front loading most of your team's income before the start of the season. If my first point isn't enough, I'll add that prices on the transfer list will go up in those weeks, and be lower markedly through the rest of the season. A bad idea all round.
I never said anything about your idea meaning more money in the game. I assumed you meant you'd just get the money at the beginning of the season, and that would be instead of, not in addition to, your gate money.
In reality, season tickets generally are cheaper than tickets for individual games. So really your idea should include a discount for season tickets. I don't really like that idea either.