Symmetry doesn't work with all tools of edit mode, besides with transform functions

(move, rotate, scale), also deleting vertex/faces/edges doesn’t seem to work either

thought, symmetrize and mirror subdtool do seem to work, but it isn’t the same thing

Yes, because you can have an asymetric model and do a symetric modelling. So it’s better to don’t have a symetrical delete for example. If you wanna have an al overall mirro then you should delete one half and use the mirror modifier.