Yeah, the op amp you drew in is just an output buffer, it’s not going to fix the problem.
Hold the phone, I have a new theory. Question: Have you looked directly at the voltage on the capacitor, that is, the voltage on the op amp pin 3? If that’s zero and you have nonzero on the output, then there’s a problem downstream of the capacitor, maybe the op amp output swing or maybe something else.
But if it’s nonzero, then check also pin 3 on the 555. That’s the voltage the cap is supposed to be discharging to. The schematic specifies TLC555CP; is that exactly what you’re using? There are different versions of the 555 and they can’t always be substituted for one another. But if I look at the TI TLC555CP datasheet it says low level output at I_OL = 1 mA is 70 mV typical, 300 mV maximum. 1 mA seems unreasonably high but they don’t quote output low level at any other value. Anyway, if the 555 pin 3 voltage is 70 mV then that’s as low as you’ll see on the output.
If that’s not it, then we’re back to thinking about the output swing. Looking at Mouser and Octopart the only suppliers I see for a chip labeled LM358N are TI and OnSemi, and both datasheets show the output swing as 5 mV typical, 20 mV maximum for Vcc = 5 V. I don’t know what datasheet @jaradical was quoting from but these look much better. So that might not be the problem. If it is, I think the TL072 with dual supply would have a chance of fixing it. But if the output swing isn’t the problem then it won’t.
If it isn’t the 555 output or the op amp output swing then my guess is it’s the other problem. You would know it if the voltage is present on the op amp input and if it does in fact go to (near) 0 V if you wait long enough, where “long enough” might be 10 or 20 or 40 seconds. In that case you might be satisfied with adding a 1 M resistor in parallel with the capacitor, as @AndreasJ suggested (and as you correctly drew), or the other fix is the “precision rectifier” solution I described above, with the diode in the feedback loop of a (dual supply) op amp.
I looked at a different envelope generator design, but which might behave similarly. This is zoomed in on the lowest 2 V, the green box is the gate, the blue curve is the unfixed circuit, the red curve uses a 1M resistor in parallel with the capacitor, and the cyan curve uses the precision rectifier. (Click on picture to expand.)
So you can see without either fix the voltage is still some tens of millivolts even after 10 seconds. With the resistor it gets to essentially 0 V at about 5 seconds, and with the precision rectifier, less than 2 seconds.
The resistor does reduce the amplitude of the envelope though:
Again this is a different circuit, and maybe your problem is something else. But those are what I’d look at.
