Don’t pool developers put the blocks together?

From what I understand, in today’s environment ASIC miners only solve math problems so that they can add the assembled blocks to the chain. It is the pool (not the hasher pool, but the developer of the pool itself) that actually assembles the blocks. These ‘pulls’ are only capable of assembling, no hashing is performed. It is the miners in the pool who actually do the hashing. Is this correct?