The reason that you don't see children's bikes for rent is the 18 year age limit to rent one.
Both for insurance reasons and the renter must have basic knowledge of Jamaican road safety laws.
And I will always offer advice to make sure everyone has the best vacation possible. If something would happen to your 7 year old by riding on the sidewalk, I would never be able to live with myself if I didn't warn about the dangers.
As Ras pointed out, I live here and have never seen a lone child riding a bike on the sidewalk. That tells me something...