There appear to be two variations on the "sailors' fashion" of shoes: one features one strap hanging loosely over the forward edge of the buckle frame. Illustrations seem to indicate that this was the more common style. The second style has the straps crossed and fed under the forward side of the buckle frame, as seen in "Watson and the Shark."
Here are some images of one of my own shoes, with the straps buckled and trained in various manners:
The buckle attached to the chape strap in the usual fashion:
The shoe buckled in the usual fashion:
The shoe buckled in the usual manner, but with the straps pulled forward:
The buckles installed on the chape strap in the conjectural “quick-release” fashion:
The tongue strap buckled:
The “quick-release” method, with both straps loose:
A possible arrangement of the straps, from the “quick-release”. (Note: this would no longer be a quick release, but may be a way to secure the shoe better, while maintaining the overall style):
“Quick release” with both straps fed under the frame:
After these experiments, I am inclined to think that my conjectural “Quick-release” method may have been used on shipboard, to allow quick removal of the shoes for running up aloft, but it seems impractical for going ashore, because it allows the buckle to release from the chape strap, too easily, and tucking the straps to make it more secure eliminates the quickness.
In all, this fashion seems to have been common from at least the mid-1760's to 1810 or so, and may continue as long as buckle shoes are common. Does it go much earlier than 1768? Further research may reveal the answer, but for now, I think we can safely say that yes, sailors did commonly wear their shoes with trained straps, for at least 50 years.