No. If the ROM was missing/unrecognized the power would stay on but the machine would not chime.
Most likely, you have leaking capacitors under the power supply in the rear right hand corner of the motherboard. There are three or four caps back there which are actually needed by the power on circuitry, IIRC, and if they leak one often gets teh symptoms you describe. If they damage the surrounding traces, the power-on circuitry can be damaged and need repair. That's what happened to my IIci back around '95.