It's a valid question, and honestly, I think they often do affect how we play. This article https://www.moviezwap.com.in/jetx-bonus-codes/ touches on that exact point  It mentions that one of the biggest changes is emotional. You feel less tense at the start, which can lead to a longer session. A longer session can be good if you're learning the game, but it can also lead to careless mistakes. Personally, I think it depends on the individual. Some people are disciplined enough to stick to their strategy regardless of the bonus, while others might get carried away.

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Having considered the whole debate, It seems like the key thing to understand is that welcome bonuses change your perception. They create the illusion of an advantage, and that illusion can strongly influence decision-making. It is probably similar to using someone else's money in any situation; you may be more willing to take risks that you wouldn't with your own funds. I wonder if the effect of a bonus depends on how big your typical bets are. A large bonus relative to your usual bet probably makes a bigger difference.

I think they can, at least for some people. Even if the bonus money isn't technically yours yet, having a larger balance on screen can make you feel more comfortable taking risks that you normally wouldn't take with your own deposit.
That said, I think the effect depends on the player. Some people treat bonuses as extra entertainment value and stick to the same strategy, while others start betting larger amounts because they feel they have a bigger cushion. That's probably why bankroll management matters so much, regardless of whether a bonus is involved or not.
So I wouldn't call it just a marketing gimmick. The psychological side of it is definitely real.