What you call Block Sub-Keys, is called key schedule.
The first thing you need to know, is that this sub-key generation needs to be deterministic (else how would you be able to decrypt the cipher text).
Thus you are considering two cases:
- a simple key schedule to generate the sub-keys.
- a Pseudo Random Generator (or more usually a Hash function) to generate the sub-keys.
In the first case you have:
$$F(K) = K_1,\\
F(K_1) = K_2,\\
F(K_2) = K_3,\\
In the second case you have:
$$H(K) = K1 || K_2 || K_3 || ... $$
where $H$ is a eXtendable-Output Functions (XOF) such as SHAKE and $||$ is the string concatenation.
In the first case you have a sort of key dependence (from $K_1$ you can deduce $K_2$, but luckily not the other way around) while on the second case each sub keys are independent (you can't deduce $K_2$ from $K_1$).
From this point of view, surely the second approach looks the safest however, one must also consider speed and memory consumption.
Generating a long stream of bits in order to have the sub-keys takes time. If you were to compare the time spend on encrypting the data provided a set of sub-key and the time to generate the set of sub-keys, with this solution the slowest one would probably be the sub-key generation. And this is not something desirable if you want to have a usable cipher. Also the second methods requires to generate all the sub-keys at the initialization of the algorithm. This requires some memory...
To sum up, I'll point you to this two answers: