If you put a slight bit of pressure on the accelerator pedal, it won't (not supposed to afaik) slam the brakes for you.
I think your system is operating as designed it's just your urban jungle environment driving style doesn't jive with it. (Not judging your driving style, I live in the midwest corn and bean fields ... I'd be scared to drive in NY City lol

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Tell me if this is a good description of the scenario that you experience all the time ...
You're driving along in traffic and the car ahead is making a right turn. You've got it all timed out, maybe even easing to the left portion of your lane a little to pass by the turning car. It may still be in your travel lane slightly as you go by but the car ahead is still turning so all is good. You've done it a thousand times. No issues. Your timing is great (46 years of driving without hitting anyone is proof enough for me!) When this scenario is happening, I imagine you are slightly on the brakes making the final small corrections to your timing and also ready to stop if things turn sour for some reason as you go on by the turning car. But now, with this Mazda, in this situation, it goes nuts with it's fast high pitched beep beep beep and rapidly flashing BRAKE BRAKE BRAKE then slams on the brakes for you. Saving you from yourself.
The problem (IMHO) is the simple system on the Mazda (and other cars as well) doesn't have the intuition of 46 years of driving experience like you do to assume the car ahead is going to continue to turn. It's simple computer box is just calculating speed and distance (and it wants to have the whole entire lane clear to pass by as well) and coming up with an answer that equals COLLISION! and responds accordingly by slamming the brakes full on.
You can simply press on the accelerator slightly in those situations and it won't slam the brakes on. That "overrides " the safety system.