I hardly ever see people fall while skating! I come from a country where lakes often freeze over in winter and, from very young, people will get on the ice. I think it's just the confidence you get from growing up with skating seen as a normal sport that means that people don't second guess themselves and so are more decisive in their movements, don't wobble & this lets them build up rhythm. Whereas when I went skating as a child in the UK, my friends were always so scared they would fall that they spent half the session clinging to the rail, which of course means you're just building up anxiety which does nothing to stabilise your body.
I am totally uncoordinated & have poor depth perception, balance, spatial awareness and all the rest of it, but skating is fine. I've skated in rinks and on lakes/rivers (where there's wind, the ice is not smooth, and there's no rail to hold onto when you need a break - long distance skating is brutal!) and managed despite a total lack of athleticism (I'll never be as elegant as my foreign relatives though, speeding along dreamily for miles with hands clasped behind backs!). I've fallen maybe three times in my life, so hardly ever. And I'm sure it can hurt but I was always struck by the fact it didn't really hurt that much, it's not like it's concrete, and you're not falling from a great height. The ice isn't as slippery as you think either.
Just force yourself off the side and after the first few minutes where it's a little bit scary, you'll find your rhythm and stop feeling it's likely you'll fall. Push first with one foot, diagonally to the side, then another, slowly, and build up speed as you gain confidence. It's honestly such a brilliant feeling when you get the hang of it.
Is your DSS happy to hold your hand? Skating while holding hands is very steadying and soo much easier, gives you confidence too.