My old landscaper told me that adding mulch around my bushes would
kill the bushes.

Hard to evaluate this claim without knowing what he had in mind. Too
thick a layer of mulch can be a problem, as the roots get buried (for
example, 12 inches a year, every year, is too much). And there are
differing schools of thought about mulches. But there's nothing about
mulching that it can only be done when the plant is first planted.

After five years I need to add more soil to the area where my bushes
are (near my patio) as the rain has washed a good amount of it away.

If the roots are being exposed, then you should definitely cover them.
Either soil or mulch would work but I'd lean towards some kind of
organic matter (composted leaves, shredded pine bark for acid loving
plants like azaleas, maybe hardwood mulch for alkaline loving plants
depending on your current soil pH), to improve the structure and
fertility of the soil. Or a mixture of soil and organic matter.

Some kind of short retaining wall (for example, a row of bricks or
cinder blocks or whatever), or some such measure, may prevent the new
soil from washing away like the old did.