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I'm using the method in a number of production designs where the hardware guys didn't provide a hardware reset signal.
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Sorry to hear you have some hardware guys around you who are really hard and unhelpful.
I myself have used internal reset instead of external one and it worked most but not every time. We assumed if the design didn't start well then the customer will instinctively power up again or give it a shake.
The power up reset was even regular reset whenever certain errors flagged that will force the counter back to zero. This also prevented compiler from optimising off the reset counter thinking that the programmer didn't really mean it.