I don't believe that is true.It may do if left in contact in a concentrated form however hydrochloric acid is used wholesale everyday by farmers cleaning miles of stainless steel piping that transports milk from the cows udders to the large stainless steel storage tanks.
Food grade stainless is not the basic cheapest 316 type that is used in secondary heat exchangers, and plain HCl wouldn't be recommended for for cleaning. There would be plenty of documentation on this, though I haven't looked.
The chloride ion goes for the grain boundaries, not stopped by the chrome oxide protective layer. Increasing Mo content helps, and is added to alloys exposed to even seawater.
The copper-alloy braze material used in cheap plate heat exchangers is also attacked. Sometimes you will see it re-plated back onto the stainless steel.
As I said, my experience, though anecdotal, suggests that the rate isn't sufficient to cause a problem. One particular hex, I have cleaned about 4 times.