The transformer supply will be definitely damaged (when there is no protection scheme) due to nearly zero resistance of the primary winding. Moreover, it is obvious that you cannot see any voltage signal in the secondary! So take the advice of licensed electrical contractors and use transformers in your AC.
But the beauty of transformers is that they allow us to have more than just one winding in either the primary or secondary side. Transformers which have more than one winding are known commonly as Multiple Winding Transformers.
Note that transformers are only used for AC systems because of the phenomena of mutual induction between the coils. We recommend you review the functionality of a transformer again and get the gist of why we use transformers in AC power systems.
Transformers, on the other hand, are able to convert levels of AC voltage and current at very high efficiencies with no moving parts, making possible the widespread distribution and use of electric power we take for granted.
In all fairness, it should be noted that motor supply sets have not necessarily been obsoleted by transformers for all applications.