Possible to insert capacitor between voltage source and output to delay the output?
For approximately 30 seconds soft start duration, there are two approaches: analog timing or digital timing.

The analog approach is to charge a capacitor through a current source (so that the capacitor terminal voltage increases linearly), and use a comparator to detect when the capacitor reaches a threshold.

A cheaper analog approach would be to charge a capacitor from a voltage source, through a resistor, then use a comparator. That’s not as good because the voltage rise is exponential instead of linear, and that makes it harder to control the timing. (Note: the buzzer should not be powered through the resistor.)

Either way, the capacitor is the biggest source of error. Capacitance will change over temperature and even change with applied voltage. Typical variation may be on the order of 80%, which is isn’t great.

The digital approach is what @Hearth is suggesting, use a microcontroller to handle the timing. If this were 1980 I’d suggest using a 555 timer or one of the related intercil timer/counter chips, but in 2019 it’s cheaper and simpler to use a microcontroller. That will still require a timing source, but a cheap tuning fork watch crystal or even the microcontroller’s internal RC oscillator should be good enough.

When you’re selecting a microcontroller for this application, some features to look for are internal oscillator, internal LDO regulator, cost of development tools and licenses (free is often possible), and how easy is it to find a community that uses and supports that chip. ARM such as STM32 are often a good choice, as are PIC / AVR.