Did some surgery to replace the old connectors on the Heathkit signal generator with BNC connectors.
I thought it might be a bit of a challenge, but turned out easier than expected. The center of the connectors turns out to be just a hole/eyelet, so a bit of heat on those center contacts to melt the solder was enough to free the wires connected to them.
Once that was done, it was a simple matter to remove the connectors and replace them with a couple of BNC connectors. Solder the wires into the center connector of the BNC and done! While I was at it, I also replaced the yellow 0.01 μF capacitor with a new one.
Looks pretty spiffy and modern now with the BNC connectors. Still not entirely sure if it works or how well it works, but at least now it’ll be easier to connect the signal generator to stuff.