Ferret man:
Low hum noise such as what you are experiencing is usually associated with either mains noise making it's way into the system, or a grounding issue, or broken shielded conductor on signal cables, or bad capacitors.
If the hum is experienced globally, then the cause must be common to everything and the power supply immediately becomes suspect because the PS is obviously common to everything. If you are only hearing the noise on one channel, then you know the power supply is OK because both channels draw from the same power supply. That is the simplest way to diagnose -- determine what is common to the problem and what is not.
So I know you said you will test the boombox on batteries but that is important. Because if you are hearing the noise on mains and not on batteries, the capacitors in your Power supply should be replaced.
Cleaning all the controls and switches may help -- especially the record bar because grounds are often switched on the record bar and poor connection due to oxidized contacts can present as a bad ground. Obviously, corroded wires, solder joints and such can cause much the same symptoms.
Then there is the shielded cables which are common to the tape deck head cables. Shielded cables typically carry low level audio signals from place to place. Low level signals are especially prone to RF interference. Insulating those signal wires with a grounded shield is a common design technique. If the shield conductor is broken or ungrounded, then the RF protection is no longer effective.
Capacitors also can fail in a system allowing unwanted noise to enter the system. Throughout the PCB, you will see capacitors all over the place. One characteristic of capacitors is their ability to pass high frequency signals to ground. This is called bypass and caps used to "bypass" is usually implemented in a circuit in parallel with a load between the supply and ground. Installed in this way, the capacitor blocks DC (allowing that signal to proceed normally through the circuit) but pass high freq ripple to ground cleaning the signal. In my mind, I'd rather replace all the caps than to test them all. The reason is that with the caps exceeding 20 years and approaching or exceeding 30 years of age, it makes very little sense to remove them from a circuit (to test) only to reinstall the old aging caps, even if testing shows that is has not failed.
I wish I could tell you specifically what and where to look but the RX-C300 is not a common boombox and has many many IC's and OP AMPS which are especially prone to ripple interference. But you are going to have to go and get your hands dirty on this one as I don't think there are any "experts" on this model. The general advice herein is the best I can do, without personally looking at it myself.
My course of action (in the following order) would be:
(1) Test unit on batteries. If hum is gone.... replace PS caps, including the small ceramic ones
(2) Thoroughly clean all controls and switches including the record bar switch
(3) Visually check for broken grounds and broken shield wire conductors
(4) Using DMM, check for continuity, especially on ground shield wires tethering different circuit boards.
(5) Consider re-capping the boombox.