When its out of frequency, the impedance is higher and thus the current is lower. When is at resonant, the impedance is at the lowest and the current is highest. It is actually here that I am a bit concern whether the design was originally catered for at resonant. But the designer did put a series resistor there to limit the current and protect the transistor.
IMO, I do not think it will burn out the board because as you have seen, the transistor is rated at 8A. At most the fuse will blow first, but since this is a "monitored tuning" by the tong meter, it won't be like catastrophic uncontrolled event. The important thing is to fill the tank with water at a level you normally will use to wash the parts and not empty.
Alternatively you can use a variac to power the isolation transformer and at every frequency steps, you slowly adjust the variac voltage up from 50V to 110vac while monitoring the tong meter not to overshoot 2A. For me, I am happy with the fuse protection and the board is so simple it didn't fear me at all. The first time I did micro smd, I was all shaky, but after that I realized that I put too much fear into it. It was all so simple and forgiving. I laugh at it while I crossed the learning hurdle.
That Heathkit IT-28 you mentioned "Green eyes" is interesting. You should also check the capacitance while you are at it. Interesting it can charge and check the leakage though I don't know how low is low leakage it can measure. I have no experience with the Heathkit. I think the bidding price is increasing after you mentioned it here.
So, we look forward to after your components have been replaced, hopefully is the component failure like you said.
Regards;
Edit: The components are aged and may not be able to be subjected to off-frequency potential. But don't you think is a good test to eliminate failing component also? Do cater for it nevertheless.