they can literally suck you into their propwash, if nothing else.
Ummm ... Sorry, Charlie. If you're that close, you are too damned close. More likely you'll get rolled & swamped by the bow wave. (I've been boating here on Puget Sound for many years, out on the bouncy water ducking in between them big fellers.) Never did hear of any little boat gettin' itself "sucked down." Now, thanks to Homeland Security, if you get within one-quarter nautical mile of any so-called HCV (passengers) or "sensitive" (oil tanker) or military (one-half nautical mile limit) you get a huge fine and time to think about it in a federal prison.
being able to refuel and restock a warship without needing to stop moving is the only reason they continue the practice
Again, there's more to it. I've been there, rode along while it got done. Gulf of Alaska waters. Two ships alongside, powered down, dead in the water are far more dangerous to each other than steaming underway alongside each other. Reason: no steerage way means no control. And rolling helplessly in them big waves means lots of bashing & crashing. So, unless it is dead calm and no current, it's better to keep under way and stay right on the helmsmen's respective asses to steer a straight course. The ships do maintain sufficient separation to allow for rolling and small heading changes. About the worst that normally happens is the poor slob getting hauled across in breeches buoy will get dunked in the middle when the lines temporarily go slack.
Ships of any size are very stable underway, despite the pitching and rolling. If any seas are running, they'll steer into it rather than wallow in the troughs. Pitching up & down is okay; rolling like a drunken sailor, not so much.