Final transistor in VOX circuit design is saturated even when no audio is supplied
Microphone Signal Level
The schematics are there to read. The microphone input looks like this:


\$R_{66}\$ and \$C_{64}\$ help isolate the circuit from the main \$+12\:\text{V}\$ power rail. Think of \$R_{66}\$ kind of like a set of bed-springs for the circuit.
\$R_{60}\$ tells me this input supports electret microphones. They pull up to about \$1\:\text{mA}\$.
The \$R_{61}\$&amp;\$R_{62}\$ divider pair look to be dragging almost \$1\:\text{mA}\$ and also setting the base to \$\lt 2\:\text{V}\$.
That now suggests that the emitter current is also going to be \$\approx 1\:\text{mA}\$. (This is looking like \$1\:\text{mA}\$ everywhere!)
That means there's a quiescent \$1\:\text{V}\$ drop across \$R_{65}\$ and that probably means that the output of this circuit should be limited to about \$\pm 500\:\text{mV}\$.
Input impedance is \$\le 1.5\:\text{k}\Omega\$. This site shows values about in that area, too. The JIS C-5502 standard has as one of its values \$600\:\Omega\$ which also just happens to be a commonly found audio transformer value, too. So about half the microphone output is expected to be tossed and I'm concluding a total gain: \$A_v\approx 6\$.
Since I know the circuit output shouldn't exceed \$\approx \pm 500\:\text{mV}\$ I can now say that the input shouldn't exceed about \$\pm 80\:\text{mV}\$. Or \$\approx -25\:\text{dBV}\$, thereabouts.

That said, the above tells me that the expected signal level isn't tiny.
(I had no choice but to look at the unit you have to see what it expects on the assumption that you have a working microphone. All that because you didn't specify your microphone details.)
I've learned something else, of course. I now know for certain that you are using a transceiver that accepts and powers electret microphones as well as accepting a relatively wide variety of other sources. I also know that the circuit you are trying to build and test does not supply power for electret microphones.
So if you are using an electret microphone that requires an external supply, then it will work just fine with your transceiver but will not work fine with the circuit you are working on.
My ignorance about your microphone has now become a problem in my trying to be of help.
VOX Module
The 741 opamp is a pretty poor choice, these days. It was designed back when only a few transistors could be affordably placed on an IC. I still have some 1458's from 40 years ago (and more) sitting in one of my boxes. That's how far back they go. But I understand why you'd resist replacing them without knowing circuit details.
This circuit works on a pretty simple principle. It includes a first stage giving \$A_v=100\$ (\$+40\:\text{dB}\$) and having \$1\:\text{k}\Omega\$ input impedance. That impedance isn't too far from the one I'm seeing for your transceiver circuit. In that sense, there's a match-up. (Of course, there's still my question about whether or not your microphone requires a small current to operate it.)
Given what I believe I understand about the transceiver's microphone input circuit, the voltage gain seems quite high. But I don't think that's necessarily a problem. It just means that the output may saturate near its rails, chattering between them, when you speak into a suitable microphone. Lower sound pressure levels may very well avoid it railing like that and instead deliver an amplified but largely undistorted result, too.
So I can't say this is a design flaw. I'd need to know more about the reasonable range of appropriate microphone SPLs and the kinds of levels of voicing that need to be included in activating the circuit. With sufficient research into appropriate microphones likely to be used with your transceiver and enough time to learn about different human voicing levels, I might be able to make a credible argument one way or another. But I'm not in the mood to even attempt that effort. So I must say, &quot;I don't know. Perhaps this voltage gain is just fine.&quot;
What I can say is that there is a potentiometer in your circuit. It has a certain range of motion providing a certain range of input voltage to one side of the second stage opamp. If this range, together with the voltage gain and the vagaries of appropriate microphones and voicing levels, allows successful use then the circuit is fine. On the other hand, if just ambient background noise and electronic noise in typical microphones happens to cause the first stage to swing around too widely, then it's just never going to work because it will always be triggering the second stage. So one possible suggestion is to consider reducing the first stage's voltage gain by changing the \$100\:\text{k}\Omega\$ resistor to a smaller value.
But that's a maybe. Again, I'm too ignorant to know for sure.
The idea of the second stage is to just operate it completely open-loop. This just means that it's output will have a very, very high gain that multiplies the difference of its two inputs. Only the smallest of differences might yield an analog output that is between its two rails. Too small of a difference to be rationally expected to happen. So without any applied input, this second stage will either be always at one rail or the other, saturated out.
The idea here is that this is okay. The output of the first opamp should be centered midway between the voltage rails. If the potentiometer sets its voltage sufficiently above or below that midpoint level, then very low level audio signals, output from the first stage, will always stay either above it or below it, meaning that the PTT is either always active or always inactive. You will want to set it so that at these very low levels the PTT is always inactive, of course. (But the circuit allows you to set it the wrong way, too.)
If the first stage gain is acceptable, then you should be able to move the potentiometer to one side or the other and get these two behaviors with very low levels of input. But if the ambient (and other) background, and/or low levels of audio that you don't want to trigger the circuit, magnified by the first stage leaves you with no possible position of the potentiometer where you can inactivate the PTT, then the voltage gain of the first stage is simply too high and you'll need to remedy that fact.
Summary
I've already suggested that you try leaving your microphone disconnected from the input and using a VOM to measure the first stage's output voltage. Although you might prefer to use an oscilloscope to see how much the noise swings the output around, I'm gathering you don't have access to one. A VOM should at least show you the average value and this should be close to midway between the rails.
Another test to try is, as I also mentioned, to use an LED and an in-series current limit resistor between the PTT output and the positive rail voltage. (Polarity correct, of course.) Without a microphone input source, you should be able to adjust the potentiometer in such a way that you can turn the LED on and also turn it off. If it stays on no matter what, then you've already determined there's a problem. Either with your wiring, parts, or else because the gain is too high and noise itself is causing continual triggering.
That's all I can recommend for now.