It all depends on the specific lenses, cameras, and adapters in question.
One obvious consideration is that an adapted lens needs to have an aperture ring, since there will be no electronic aperture control. Also, there will be no auto-focus, obviously.
This is not as obvious as you seem to think it is. Many adapters do translate the aperture control and AF protocol from the camera's system to the lens' system with varying degrees of success. Some even do it very well.
Remember, all micro four-thirds cameras use an all-electronic connection between the camera and lens. Many photographers who adapt other lenses to micro four-thirds cameras choose Canon EF lenses precisely because the EOS system also uses all electronic communication between the camera and lens. This means there is no mechanical connection(s) to try and replicate when using EOS lenses adapted on µ4/3 - just data signals that must be translated.
So, given these constraints let's imagine trying a lens like the Sigma 70mm F/2.8 macro. This lens is renowned for sharpness. So, if we mount it on an MFT will it be even more sharp and distortion free because we are only using the center of the glass?
In the case of the Sigma 70mm f/2.8 Macro you probably won't 'gain' much, if any, sharpness using it on a cropped sensor. Why? Because the primary reason that particular lens (and other flat field lenses) are considered 'sharper' than some of their counterparts with a more curved field of focus is because they remain sharper to the edge of the field when imaging a flat target. Lenses with field curvature can be just as sharp at the edges as a flat field lens can, they just can't be as sharp in both the center and the edges at the same focus distance when imaging a flat test target.
Will this extra sharpness be noticeable?
What extra sharpness? See above.
An additional caveat is that the field of view will narrow by a factor of 2, so the Sigma will shoot, I guess, like a 140mm in terms of crop factor and magnification.
That may be considered a caveat or an advantage. It all depends on whether you'd rather obtain a narrower or wider FoV with any specific lens. This is exactly the same as non-adapted FF lenses used on APS-C cameras with the same native mount. Birders love the 300mm equivalent FoV obtainable using a 200mm lens on a 1.5X APS-C crop body. Landscape photographers don't like the 36mm equivalent FoV obtained with a 24mm lens on the same APS-C camera near as much (or at all).
This entire issue can be compensated for with a speed booster that effectively changes the optical power of the lens in the reverse manner to the way a teleconverter does.
There are additional considerations.
- Adding an adapter introduces an additional mechanical interface that can introduce small alignment errors between the optical axis of the lens and the plane of the camera's sensor. Variations of as little as 20 microns in flange distance from one side of the interface to the other can be seen in images made using fast wide angle lenses.
- Lenses designed in the digital age usually account for the thickness of the filter stack in front of the image sensor for the native mount system for which they are designed. The total stack thickness varies from one mount system to the next. This can affect the overall optical performance of the lens, particularly with regard to maximum sharpness. This is why Roger Cicala and his crew at lensrentals.com/OLAF now test lenses with a cover plate of the designated thickness for each lens mount system (or specific camera) on their optical bench to simulate the filter stack when testing lenses for a certain mount system.