Why are multiple rounds with generated subkeys used?
It's true that a badly designed key schedule weakens the cipher.

However the main point is to "stretch" the randomness across all subkeys by using diffusion techniques, while using only $k$ bits of randomness for the key instead of $kr$ bits of randomness if all round keys were independently chosen. 

This is doable. AES is a good example where the components of the cipher are used in the key schedule.

More theoretically, James Massey (and one of his students whose name escapes me now) has proved that a construction called perfect local randomiser can extract random looking $k-$tuples (subkey bits) from a uniformly distributed $n-$tuple (randomly uniformly chosen master key) such that these $k-$tuples are locally uniformly distributed. The construction uses orthogonal arrays which can be obtained from certain MDS codes under certain conditions. MDS codes are intimately related to the branch number of AES MixColumns operation, see, e.g. The answer to this question.

Edit: Also, a one round cipher will not have enough diffusion for linear and differential probabilities to drop to safely low levels, as it happens through multiple rounds.