My 4 MPH mobility scooter has a 270 watt motor, and I know my mother was able to run into me without really causing any harm, I stood there to stop her hitting some one else. However I have read reports of accidents with mobility scooters, and one wonders how they managed to cause so much damage.
The problem with mother was compounded by poor eye sight and I know in Rhyl see drove off a curb which she though was a pattern in the pavement, I read about
this accident and my thought was yes could see my mother doing the same, the guy likely did not realise there was glass, but true accidents happen because of some thing we have not foreseen, and with any home made item it is hard to work out what might happen.
However judging from size of mobility scooter motor which is designed for an over weight adult, I would say 800 watt is too large. The scooter weighs 67 kg and uses 2 x 35 Ah AGM lead acid batteries, range 25 miles. I hope this gives you some idea was what is needed, max user weight 150 kg. As already said 270 watt motor, rated 16 amp, 3,300 RPM, 8 kg/cm torque, and of course 24 volt.
Your unlikely to build such an efficient unit, but it should give you some pointers. I would guess it uses some pulse width modulated controller which is not really some thing your likely to build at home.
We would all have liked to have built some thing like that for our children, however I did not have the space they could have used it in. May be this social distancing will cause some relaxation in rules for electric powered vehicles, seems mobility scooters are only permitted where you can't walk, electric bikes have a 16 MPH cut off for assistance 14 year old age limit, 250 watt power limit, must be only pedal assist, at least two wheels with brakes, etc. This has resulted in one wheel and scooters all being not allowed on British public places, and the problem with e-bike is not easy to carry on bus or train. So we have all seen the young child with the electric car, and to be far some have a remote for parents so they can remotely switch it off, so you can't really claim not safe, but when restricted to private ground it means they are rare, so nothing much to copy from.