Hmmm... I think it's still stricter to consider Science a philosophy than the other way around. It's the belief (and an extremely useful and successful one) that the nature of the world can be understood through observation, experiment and deducing mathematical relationships between things. There branches of philosophy that are not strictly scientific, but nothing in Science that is doesn't rely on the fundamental philosophical principle of empiricism.
But we use the scientific method via philosophical inquiry, so I think it comes down to how we decide to strictly define these things. I definitely agree that certain definitions lead to the same logic you've presented.
I think this is backwards, no? Science is a philosophy, not the other way around.