I do think I am fortunate that I have lovely families who trust me implicitly and from the very beginning when I have taken sole charge of their precious children (youngest I've had are 7 week old twins although I've had younger siblings but they knew me well by then as I'd had their older child for a few years) while they go back to work full time.
My families respect me and treat me as they would want to be treated by their employer. I get sick pay (but then I never take the piss with sick days, have been with my current family 3.5 years and haven't had a single day off sick. In 32 years I have had 11 days off, which is stated in my references)
I get all my holidays paid, even if they go over their allowance.
They are very rarely late but they let me know the second they know they are going to be late, not at the time they are due home.
Most of my charges grandparents have walked in, said give me my grandchildren and go and take a break until xx time, I still get paid for that time. My bosses wouldn't even think not to pay me.
I think you need to refine your interviewing technique. Remember you are interviewing them as well as them interviewing you. Ask what would happen in the scenarios you have posted about and if they say if you don't work we don't pay you, you say thanks for your time but your family is not what I'm looking for and leave.
And make sure you have a water tight contract that says they pay you for your contracted hours even if you are not needed. That they will pay you on time or they pay any bank/company fines that you get for having insuffient funds (mine also states £25 a day late charges for the inconvenience of having to call everyone and make another payment, not because I've ever not been paid on time but because I've read/heard stories like yours and I won't take the chance.
If you have a good track record for illness, tell them you have never taken the piss and that you'd like xx days at full pay and xx days at statutory pay.