FPS counters in games usually display an average across multiple frames. That makes the number actually legible if the FPS fluctuates, but if it fluctuates really hard on a frame-by-frame, it might seem inaccurate. If I have a few frames here that were outputted at 20 FPS, and a few there that were at 70 FPS, the average of those would be 45 FPS. However, you could still very much tell that the framerate was either very low or very high, which would be perceived as stutter. Your aforementioned old games probably were frame-capped to 20, while still having lots of processing headroom to spare for more intensive scenes.