No one remembers anyone but the man who wins. I want to be remembered, so I focus on winning. No one cares about second best, and neither do you or myself. I only ever focus on people who win - Bruce Lee for example.
Never implied that to "not think of winning," one cannot win. You are reading into Funakoshi's quote a surface conclusion and perhaps doing him a disservice. I'm sure he did not believe in being second best. Go deeper into what that quote can mean.
Most of us have experienced driving home, and got there, without actually being aware of driving there (even when sober.) In kyudo archery, not thinking of the target is a key concept. Taoist philosophy teaches similar ideas. (Bruce Lee was very familiar with Taoism.)
If one adheres to proper martial art principles, has strong mind, body and spirit, and skillfully executes techniques, the win will come by itself, naturally. Sometimes trying too hard, focusing too much on the "win," takes away from the path to get there. Older, more experienced, martial artists may have a better appreciation of what Funakoshi was hinting at.
Like walking on rough trail, bent on your destination, eyes glued to the warm cabin up ahead, only to trip on a branch and break your ankle, or step on a snake and get bit - you are "lost." Should have been more concerned with the path, executing its route, and have been thinking of not losing rather than the win of getting to the cabin. (Yes, I love metaphors.)
A fundamental military doctrine, as true now as a thousand years ago, is that before setting off to battle, the following must be accomplished: Lines of supply and communication must be established and defended, flanks and rear must be secured, and the ground be to your advantage. This insures that you are not vulnerable to the enemy's attack. So, a good general 'thinks of not losing," this being the second part of Funakoshi's quote. Then, the general's army, employing good training and tactics, fitness, and courage (mind, body and spirit) will most likely be victorious.
That's my take on the A part of my post. The B part may be just as valid. Even so, can both views co-exist? Or, as I see jpseymour just posted, each has a separate purpose?