Simply handing drug dealers and corrupt politicians a boatload of money isn't likely to do much of anything - he'd be bankrupt in a year and the city worse off than when he started. That's why the Harvey Dent arc was so crucial: Batman can only do so much in the shadows, but what the city really NEEDED was a hero who could operate in the light of day (though he still needed support from the shadows).

Ofc the real answer is that the premise of the franchise is based on Batman punching people, as in physically, so his goal isn't even saving the city so much as making satisfying wham bam pow sounds.

More "political" franchises are fewer and further between, which is why Star Wars and to a lesser degree Trek (in this regard) were so popular. Both involved a radical, violent and bloody overthrow of the corrupt forces (Trek having been in the past but in Wars it happening "live" and being the central feature).