Good points ZephroCarnelian. The bow and arrow point is an argument in favor of gadgets.
I think that Spider Man can fight very naturally and achieve the martial arts effects you mention because he has the instincts of a spider and superhuman reflexes and strength.
Do you guys think there is a limit to the amount of pure strength, speed, and agility that a peak (but not superpowered) human can take on? That is basically my argument here regarding hand to hand combat. Personally, I think that there is a limit. I believe that Spider Man is beyond that limit for Batman.
Gadgets are another matter. There is a threshold of technology beyond which Spider Man's abilities won't help him anymore. I think on average Batman's gadgets don't pass this threshold.
So it's obvious what's going on here. You think that Spider Man's power level is not great enough to take Batman, and you also think Batman's gadget level is too great for Spider Man. Fair enough. Maybe it is possible to find examples of both cases. Probably depends on the writer. But based on what I have read I strongly believe in what I wrote above, and part of my opinion is based on the nature of the characters (one superpowered, one not).
It isn't that I don't understand basic ideas of martial arts, or the meaning of the word agility, or that technology can be powerful. I just think that a human has limits and a superhuman has higher limits in the areas that makes him or her super. This belief gives Spider man the advantage in my opinion.
I don't want to take away from Batman or disrespect your opinion. I just believe in mine based on what I have read in comics.