No, it is not necessary to stand on it, and that would be utterly dangerous. Just squat on it directly, first one foot, then the other. Be careful when getting on and off it, using something for support if available. It is not for people with a weak balance. Position as forward as possible so you land correctly. It is a strange thing to have to do, but it will fix and prevent numerous associated issues.
It will lead to more splashing, so use the flush a lot more. It can also help to take a shower thereafter to keep the legs clean.
A side benefit is that no one will want to remain in this position any longer than necessary since it tires the feet, so you don't waste much more time than you need.
I have an appropriate stool, and have used and compared it over a lengthy period. The subjective benefit over time is not 90%; it is about 30%. When squatting, there is no risk of falling if one has normal balance and is not unwell. It is not for the inattentive person. It takes weeks or months of use to realize its pelvic benefits.
The risk of falling is highest when getting off the seat, at which time it helps to have a support to hold.
The correct solution would of course be to remove this toilet altogether, replacing it with a squat toilet.
The noted actions are not something that anyone will typically do until they feel they're out of options, but then it's a potential option.
I do not understand why you are downvoted. My parents came from a country with squat toilets, so they taught me to squat on the toilet seat, and I have gone that way ever since I was toddler (except in public restrooms).
It is very inappropriate to do at a workplace toilet. If there is excrement on the toilet seat, they're not positioning themselves correctly, or are too large for the seat, and they don't care to cleanup either. Wearing shoes alone is a reason to not do it outside of home.
It will lead to more splashing, so use the flush a lot more. It can also help to take a shower thereafter to keep the legs clean.
A side benefit is that no one will want to remain in this position any longer than necessary since it tires the feet, so you don't waste much more time than you need.