You need to find a mechanic with a cooling system pressurising tool. What that does is pressures the cooling system when engine is cold simulating the conditions when the system is operating. Applying about 1 bar of pressure should reveal the point of leakage.
This is what I am getting at: https://youtu.be/jcEow4CC2ME?si=aBp2uERLOw-7ohsn
From what you have said about the questionable condition of thr radiator, I am guessing that the previous owner has been using water instead of approved Honda coolant. I had a gen X for 6 years with factory fresh cooling system even after 6 years as the car always had Honda coolant in it. BTW coolant life is 3-5 years.