Update: we now have a Panasonic 200mm f/2.8 and Olympus 300mm f/4 lens to add to this discussion, but the same things apply.
Wildlife and bird photographers are always searching for smaller, lighter, cheaper solutions with more reach. The problem is that when you change one parameter you change others, as well. Put another way: there's no perfect answer to what you seek.
The plus of the 100-300mm on an m4/3 body is that you have a very narrow angle of view, about 3.5 degrees horizontally. That's equivalent to a 600mm lens on an FX, A7, or 35mm film body. You get that in a package that's smaller, lighter, and cheaper than any 600mm solution for the larger sensor bodies. The more recent 100-400mm nets you an 800mm equivalent.
The minuses are these: first, you don't have a lot of mass and the lens is small with a very close-in rotation point. It's a bit tough to handhold 300mm on m4/3 well, even with image stabilization. The f/5.6 maximum aperture and m4/3's relatively rough high ISO capability put you into a problem area with shutter speeds, too, especially in edge-of-the-day lighting or deep shadow areas. It's possible to get "stable" shots, but it's definitely not easy nor is it guaranteed that you can get them. Note also that the 100-300mm does not have a tripod collar, which can become another problem if you try to use a tripod to smooth things out, especially a tripod with any jitter to it. (The 100-400mm has a tripod collar.)
But the bigger problem is that f/5.6 aperture (or f/6.3 on the 100-400mm at 400mm). Given the smaller sensor size, you're getting a shot that looks as if you were on a full frame camera with f/11 (f/13). That's actually three full stops slower than the lenses that the pros use to get those great wildlife images, so it means you're going to have a tougher time isolating animals from the background due to the extra depth of field.
So I guess my answer is mostly centered on your use of the words "good" and "replacement" in the question. In my experience, no, an OM-D (or other m4/3 body) with a Panasonic 100-300mm is not a good replacement for something like a D7500/D500 with a 400mm f/2.8 lens or a D750/D850/D5 with a 600m f/4 lens. You'd be fighting a few issues that you wouldn't with those bigger cameras and lenses. That's not to say you can't get good or interesting shots with the combination, but you'll be fighting a lot of issues to get there. Is that worth doing to get smaller, lighter, and cheaper? Not in my book.
On the other hand, if an m4/3 body is all you've got, then the 100-300mm or 100-400mm is the most versatile of the long telephoto bunch as I write this. Just be aware that you don't always get something for nothing.
