Obtain all possible Turing Machines from a Universal Turing Machine
Enumerating all the possible TMs is a boring programming exercise. We do not need to use the UTM at all here -- we only have to generate the representation of all TMs and output it, using the chosen encoding of TMs as words.

The UTM does not accept your language $L$ since it accepts another language, namely $\{\langle M,w\rangle \ |\ \mbox{$M$ accepts $w$}\}$.