What you deposit is already intended for gambling, so whether you go all-in or divide it into smaller bets, the reality is that bankroll management won’t change the odds. It just adds an extra layer of stress in making decisions that, in the end, won’t really affect the outcome of a luck-based game.
I disagree. You still need bankroll management. It's true that the money we've deposited in our account balances is intended for gambling. The situation might change in an instance that we're not expecting.
Some unforeseen events could happen, and that money is being needed for that purpose, so we can withdraw it anytime we wish to. Or, if we're in a losing streak, doesn't mean that we can let all of it lose.
The both of you have two different right answers on the subject, but bankroll management is the application of responsible gambling. The funds could get finished, applying this strategy, but not as quickly as that of a player who goes all in. Delaying the end of a bankroll, ensures longer duration of gaming; a player could gamble 5 hours a day and save his bankroll for 3 weeks, including subsequent wins. Hence, for a gambler who understands the house edge it's of no essence to finish it immediately. It'll also cost more money to replenish the balance in a fresh session.