Previously, I used to use bigger reset capacitor and resistor to get longer reset times. How does this method fare compared to a brown-out detector as a Vcc rise time solution? I stopped it because of the common belief that a bigger capacitor can harm the reset circuitry since it is capable of discharging through the reset pin. Please comment.

it is NOT the capacitor value, it is the fact of using a slowly rising voltage to a logical input (the reset pin). Also, you get no reset at power down, and that will cause runaway at that time.

Why bungle along trying to change what does not work to something else that does not work, there are supervisor chips available for less than $1.

a RC reset is (usually) good enough for fast rising Vcc but always will fail miserably when the voltage drops (error or power off)

Erik