Which PRNG security requirements do I need for key generation
Your idea is not bad, but not a magic bullet. Any serious vulnerability at the root CSPRNG as you describe is fatal for the system facing any serious attacker.

A serious vulnerability being that the attacker can at will force values or predict (partially) values generated in a particular timeframe.

A serious attacker is an attacker knowing your protocols and schemes - if you're worried about things like this and follow Kerckhoff's principle then you should assume you are indeed facing a serious attacker.

However, I doubt that things like CryptGenRandom face a serious vulnerability as described above. This does however not mean it's immune to things like related key attacks and biases. This is where your idea could potentially help.

If you're slightly distrustful of the underlying RNG and need high volumes, hash it and most potential biases/attacks should disappear.

If you don't require large volumes of seeding data, get a lot of output from the RNG and use a KDF to limit key space attacks.

If you want any kind of security that's worth it's salt get a hardware random number generator.