I think on reflection, my comment earlier was a tad harsh. But it sounds from OP like it wasn't a terribly serious injury and OP's DS had a support network very quickly available. (By not serious I mean, didn't hurtle down three flights, wasn't unconscious and didn't have bone sticking out etc etc)....
That said, no one should ever feel deterred from calling for help when needing it. However, DH (paramedic), spends his days dashing out to people who are at best 'slightly unwell'. He also picks a lot of old ladies up off the floor, and he also meets a lot of very sad and lonely people, who while not suffering from a life threatening emergency, in that moment, just really need someone just to be there.
He hugs people, he cleans shit off them and himself, he reassures, and very occasionally he comes home really excited that he went out to someone at deaths door - not because they were at deaths door, but because that day, he genuinely saved a life. He is busy every minute of his 12 hour shifts, but it is rare that he actually does save a life.
I do think we should always consider whether we really need that ambulance - are we calling for convenience, or because we are genuinely in trouble? I think plenty of the people who call could make that choice rationally and sensibly. (Some people just call for the drama too - he sees that a lot).
The system isn't perfect, but on the one hand our colleagues in the NHS are battered and complained about, and moaned for not attending quick enough, and the other, a lot of us do take a bit of a liberty with what we expect to be provided.
A lot said on this trail - but hats off to our hard working NHS staff who are always on the brunt of it all - whether from patients or targets or tabloid outrage.....