It IS possible to get a poodle mix where both parents have been health tested, you just have to be willing to find the right breeder.
I will say though, poodle mixes take a lot of work. Generally, they don't shed (not a given, but if they are a few generations in it's a lot less likely), but because of the combination of coats they are very prone to matting.
I have a large breed poodle mix, and have taken him to the groomers every 5 weeks since he was 12 weeks old (so the last 7 months). He also needs to be brushed daily, and washing him post rainy/muddy walk is an absolute pain in the arse. That being said, it is our choice to keep him long of coat - he's very fleecy/wavey, if we just got him clipped short our life would be much easier. Our groomer said that poodle mixes are her bread and butter, but she doesn't have a single client with a dog our size that keeps them long and deals with the upkeep like we do.
Yes-they are everywhere, yes- there are breeders that could not give a shite as long as they make some money on them and yes - I'm sure there are a few out there that have picked up the worse traits from each breed, but that could happen to any mixed breed dog. There are some really, really badly bred pure bred dogs, it just seems that poodle mixes are as popular to slag off as they are to buy.
Puppies are hideously hard work, and take a lot of time and effort. Wait until your child is primary school age at least, and then do your research on breeders.