I can best ask this by setting up a situation: A friend and I are standing in the same place pointing our camera’s at, say, the same bird that is some distance away. I have a micro four-thirds crop lens, he has a full frame lense. The bird fits nicely into both images (with different fields of view of course), when each camera has a actual 200 mm lens. I know the effective aperture is also affected, but my question is: Does “double the effective focus length” mean anything other than change the field of view in this situation? That is, does doubling the effective focus give the 4/3 lens a greater “reach” or improve the focus or improve the ability to “see” the bird? Or is all that happens the change in field of view? Put another way, as the bird moves further away and we both try to follow it with cameras, will the crop sensor be able to see it more clearly at greater distances?
Does “double the effective focus length” mean anything other than change the field of view in this situation?
...is all that happens the change in field of view?
...will the crop sensor be able to see it more clearly at greater distances?
Nope, not really. It depends both on the resolving power of the two lenses, and the resolution/pixel density of the two sensors. But if the pixel density was the same, and you cropped the full frame data down to the four-thirds area, they'd be the same.
In general practice, the smaller sensor and lens typically won't resolve as well as an equivalent longer lens on a larger sensor would. That is, if your friend was shooting with a 400mm lens on their full frame sensor, it's likely they would have a better view of the bird than you would with the so-called equivalent 200mm on micro four-thirds, even though the FoV/framing would be close to identical (I hedge because focal lengths on lenses are rarely 100% accurate).
Superzoom bridge cameras with 1/2.3" sensors often demonstrate this. While their focal length equivalency is mega-boosted by the 5.6x crop factor, a 100mm lens still resolves and magnifies like a 100mm lens, not a 560mm lens. :D
See this image and explanation of crop factor from dpreview's Canon 5D (classic) review, which demonstrates what's really going on.
With the same 200 mm focal length lens on both cameras, the lens projects the same image of the bird... Same size image of just the bird in both cameras (because both are a 200 mm lens). It does not depend on the sensor, a 200 mm lens does what 200 mm lenses do.
The only difference is that the camera with larger sensor sees a larger frame (a larger field of view) around the bird, and the smaller "cropped" sensor size crops that frame, sees a smaller frame (smaller field of view) around the bird. The same size bird image does more completely fill the smaller sensor frame. The full frame sensor sees a frame with 2x dimensions than what the 2x cropped sensor sees (different fields of view, if with the SAME lens), but with the same lens, both see the same size bird inside whatever frame.
The sensor size absolutely CANNOT MODIFY what any lens does (instead, it is as described above).
So I prefer the word Equivalent focal length rather than Effective focal length. Effective suggests that the lens does change magically somehow to see something different, but a lens never does change (unless you zoom the actual lens).
Equivalent focal length means two situations (sensors and lenses) matched so that both both cameras see the same field of view, meaning the framed size of the field of view. The Four-Thirds sensor has a 2x crop factor. The 1x full frame sensor has to use a lens 2x longer than a 2x sensor to see the Equivalent field of view. In this case, the bird will be half size on the 2x camera, due to the 100 mm lens. However, the smaller 2x cropped sensor is half size and has to be enlarged 2x more to be the same viewing size (same 6x4 inch print size for example). The frames field of view appears the same then, but the bird does appear 2x larger (in that half size frame) if enlarged 2x. This is simply the extra enlargement when we look at the enlarged viewing size. So only because the smaller sensor has to be enlarged more (to the same viewing size), it does appear zoomed in. However, it is enlarged 2x more.
You see exactly the same effect if you simply look at ANY existing image in your photo editor, and then zoom in to see it 2x size but cropped to half field of view size. You should try doing that to see the exact same concept, for exact same reasons. Then it does look like a 2x longer telephoto lens was used, but the 2x enlargement is what does it. I have tried showing this effect at my site at https://www.scantips.com/lights/cropfactor.html . The small sensor also crops it to half size, and the extra enlargement enlarges it 2x more, same thing. This zoom would be a "with same lens" example (not Equivalent field of view), but the bird in it is 2x size (because it is enlarged 2x to view same size print frame). A lens always simply only does what the lens does, which is not affected by the sensor size. The only effect is the smaller crop and the greater enlargement.
So yes, the necessary extra enlargement of the smaller sensor creates the illusion of zooming in, seeing it like using a longer lens on the larger sensor (just like you see when zooming in your photo editor). The smaller sensor is a smaller image, which crops a smaller field of view with a shorter lens which can match the field of view that a longer lens would see on a larger senor. Then the subsequent 2x viewing enlargement has the effect that the smaller sensor does seem to magnify the image, but it is the resulting greater viewing enlargement that does it all.
The Equivalent focal length is NOT mounted on the smaller sensor, its lens is of course always its real focal length. The Equivalent focal length is mounted on the larger sensor to see an "equivalent field of view" of the smaller sensor and lens.
And this apparent 2x zoom does come at the cost of requiring 2x greater viewing size enlargement.
So... your "able to see it more clearly at greater distances" is a debate. Yes, sports and wildlife photographers might prefer the cropped sensor, imagining it does somehow magically magnify their lens view of the bird (only after they enlarge the smaller image more to view it). But in fact, 2x more enlargement reduces resolution to half of the dpi number, so more "clearly" becomes questionable if using Equivalent lenses. But if using the same focal length, then the bird image should be the same size, and if there are enough pixels, it is a choice if to enlarge it more or not (which is of course the same choice then on either sensor).