Changing stems can definitely change how a bike handles because it affects weight distribution and how your hands interact with the steering.

However, for the change the OP is looking at, I don’t think it would be noticeable. What would be noticable is the feel of the fit.

It is true that a stem change will not affect how the bike steers no-handed. So if you don’t ride with hands on the bars, it will not make a difference.