If your priority is to get a given card in your starting hand, worst case scenario, you go first and keep 3. You can replace all 3. If you have a particular card you want in your starting hand.
36,6 % of drawing a given card in your starting hand if you go first, and
46,9 % of drawing a given card in your starting hand if you go second.
This is assuming you have 2 copies of that card in your deck. Apply this to every card you're mulliganing for (so, around 4 cards I guess) and it's highly likely that if you mulligan for them, you'll have at least one card from your mulligan every game, with some negative variance here and there.
So you actually have plenty of control. A good mulligan is super important.