So I'm a photographer that also happens to use drones for my business. While all the basic rules still apply to drone cameras, in practice variable apertures are used different on drones than on DSLR or mirrorless cameras.
Firstly, yes there is some effect on the overall sharpness based on aperture. Usually when shooting wide open the image will be a little softer around the edges, and as you stop down 1-2 stops that will typically go away. Given that most drones are using wide-angle lenses, this can be apparent, though usually not to a great extent.
Another reason a DSLR or mirrorless user will vary their aperture is to control depth of field. However, sensor size, and distance to the subject often have more effect on depth of field than aperture, and in most drone shots, you're using a smaller sensor, and are far away from your subject, so a narrow depth of field usually isn't the objective.
The main reason, in my opinion, that a drone user would want a variable aperture is for controlling exposure. Especially when shooting video, you're typically using a fixed shutter speed, as low an ISO as you can manage, and that only really leaves you two options left for controlling exposure: Aperture and ND filters. Now videographers using their mirrorless or cinema cameras obviously go for ND/VND filters, because they can and do make use of depth of field, but also they can reach up and adjust or swap filters very easily. That is not so easily done with a drone, if you've got an ND16 on to get the perfect exposure and you fly out and start capturing video, and cloud cover rolls in and now your shot is too dark, you have to bring the drone back in, land, swap for an ND8 or adjust your VND, take off again and return to where you were filming, only for the clouds to blow away and exposure to change once more. Obviously you have some flexibility with ISO and such, but in general it is much easier to just adjust aperture mid-flight to adjust for changing lighting conditions.
This doesn't mean you can't do it with a fixed aperture drone, it just makes it less efficient and more time consuming. A "pro" drone pilot will want a variable aperture as to them their time is money, whereas a hobbyist may be perfectly happy to spend twice as much time to get the shot just right. It's all about having options.