First, I want to thank you for including the number of the circuit board, PC406. This is usually a big help.
Just one problem. That number does not show up on the info I have for the original 1977 President Grant 40-channel radio. That radio has a 4-pin mike socket. The later version from the early 1980s has a 5-pin mike socket.
I'm fairly sure the one you have uses the large uPD858 PLL chip. This would make this radio about 40 years old. It's very common to find failed electrolytic capacitors in a radio that old. They can cause odd failures like this.
I would first check C59, a 47uf electrolytic cap. If it shorts, this will disable the SSB-only mike-audio amplifier TR16. Would not disturb other transmit functions. A voltmeter on the positive side of this cap should show around 7 Volts DC while in transmit for either USB or LSB, but NOT for AM.
Also, C58, a 1uf electrolytic cap will cause this if it becomes an open circuit. If it does, the mike audio from TR16 won't reach the SSB modulator chip IC2.
A radio this old is a candidate for having ALL the electrolytic capacitors replaced. Common name for this is to "re-cap" the radio.
I don't recommend this unless you're fairly confident about a project that ambitious. Eventually these radios will become old enough to make that remedy necessary, rather than just desirable.
73