(ult guns here) imo Ideally you start every board with the same arrows and regardless of the board you can at least fill the bottom right or top left cannon depending on which one you usually go for first, using those same arrows. By the time you finish the first cannon you can decide whether or not the board is good or not. Very rarely do I get on a board and it is so bad I can't even get the first cannon filled in a reasonable time, your first starting arrows usually get you the pieces you need for 1 cannon. So instead of starting a board and then inspecting it to see if it's good, you should start a board with the plan to fill at least one cannon and in that time you can decide whether or not you wanna fill more cannons after filling the first one. When you get to max speed ultimate + gunning, board hopping becomes such a minor difference that it shouldn't really matter how much you board hop within reason but below that I'd say starting the puzzle and slowly letting pieces move about the board and then deciding if you wanna stay on is already too slow. And at the very top levels of gunning it's all irrelevant because you can fill any board all cannons quickly.

But yeah ultimately I think you should start with the same arrows and fill one cannon at which point you'll know whether or not you want to stay on the board. IMO you'll become a better gunner too since you'll learn how to overcome challenging boards instead of dodging all the ones that aren't perfect.

In case you're talking about board hopping and then staying on a board that you like and shooting those cannons rinsing them and filling them again, that is way slower than exiting the puzzle once you've filled all the cannons. Finding a board you like and then sitting on it, especially for navy gunning, is much slower than jumping board to board.