A cog is also fairly slow and not very maneuverable, neither of which seem to be good qualities for a smugglers' ship. Being flat-bottomed also means it can't go upwind by sail and therefore has to be rowed, meaning a need for more crew as rowers.
If you're familiar with the TV series Black Sails*, set in the early 1700s, you'll note that the ships (all keeled) never dock at Nassau but instead anchor off and row their (usually pirated) cargoes ashore even in daylight, I suspect because Nassau isn't (or isn't portrayed as) a deep-water harbour.
This is how I assume smugglers at night would work as well, as a small boat or dory is easier to conceal than a full-size cog (or whatever) and quieter to operate.
* - and if you're not familiar with it, get yer mitts on it and watch it - it's excellent!