How do I improve sharpness of foliage and distant objects during video recording?
You are most likely observing Optical Aberration - see https://en.wikipedia.org/wiki/Optical_aberration and also https://en.wikipedia.org/wiki/Defocus_aberration. Haze is also likely a contributing factor, see https://photo.stackexchange.com/questions/93585/are-there-tools-used-to-remove-heat-haze-or-other-atmospheric-conditions-for-lon.
The further back you are from the subject, the more light dispersion / scattering you're going to be getting from the subject. Closing the aperture as much as possible will help (so long as there's enough light remaining for the sensor to get an image), if you test this, you should be able to see visible differences in sharpness between aperture fully open and aperture almost closed.
Haze you can probably do less about, you're basically talking about scattering of light due to imperfections in the air - adding more distance between you and the subject is also adding more imperfection. The stack exchange post above suggests use of different filters to assist, that's not something I've tried. I mostly film theatre productions indoors, and there's a definite image quality difference the closer I can get to the stage. But usually I'm stuck at the back (the show is primarily for an audience, not the video guy), and everyone accepts that shooting an extra 20m to 40m through haze in a theatre is going to result in an image that isn't quite as sharp as what we could get up close.