Using auto ISO is an option, but with any kind of auto exposure the exposure can change as you move the camera around. The exposure shift is reactive - it happens after something brighter or darker comes into the frame. To me this screams "amateur". Compensating for auto exposure in post is not easy.

Most zoom lenses are variable aperture. It's not just a question of the maximum aperture (smallest f-number setting). Regardless of the aperture setting, the f-number changes as the lens is zoomed, making the image brighter or darker.

Some constant aperture zoom lenses are not constant near the maximum aperture setting: the f-number gets larger as you zoom in. But they're still considered constant aperture. Most large aperture TV zoom lenses with long zoom ratios are like this.

The advantage of a constant aperture lens is, just as you said, so that the exposure does not change as you zoom in or out. If you zoom in or out during a shot, maintaining constant brightness is critical for having the footage look good. If you only zoom between shots, having a constant f-number is merely a convenience so that you don't need to adjust the shutter speed, aperture setting, or ISO setting to compensate and try to maintain a constant exposure.

As for performing exposure compensation in post: In a standard video gamma color space, you can't do exposure compensation. In linear color spaces like raw and ACES, you can and it's not difficult. But it's much easier to compensate for the changing brightness of your scene than it is to compensate for the exposure shift of a variable aperture zoom lens.

I have piles of constant aperture zoom lenses. Even when I don't zoom during shots, the convenience of not having to adjust the exposure is tremendous. I also have variable aperture zoom lenses which I only use for photos. Most of my photo shooting uses auto ISO, so the exposure shift is not so much of an issue.

The depth of the focus field is a function of several variables. It would be very difficult to make a zoom lens with constant depth of field, and I don't think any lens has this. I'm not sure it would be desirable even if it were possible. For all zoom lenses, depth of field decreases as you zoom in.

Making a zoom lens have a constant aperture is a design trade-off. To maintain a constant f-number, the entrance pupil diameter must increase with the focal length. Achieving that usually makes a lens larger and heavier than it would be if it were variable aperture. That's why photographic zoom lenses are usually not constant aperture. But TV zoom lenses are almost all constant aperture.

Personally I prefer a software zoom over a mechanical zoom if you aren’t zooming very far and you have 4K footage that will be down sampled to 1080p anyway. However, I know that isn’t always the case.

Have you tried the Auto ISO with the GH4 for Video? I will admit that I don’t really zoom that much in video so I haven’t fully tested it yet. However, I think you will find that it is actually quite good for maintaining exposure smoothly.

In my experience the micro adjustments that the camera does because the lens isn’t parfocal are much more distracting than the micro exposure changes.

Another thing is that I don’t ever deal with controlled lighting. I am always shooting in inconsistent lighting run and gun style. Manual exposure is unrealistic in those situations because it is constantly changing. Auto ISO is perfect though. I guess I have just become accustomed to it. However, I think Auto ISO in the GH4 works very well.