Franklin is talking about Philadelphia having no paved streets:
the rest of the street not being pav’d, whenever a carriage came out of the mud upon this pavement, it shook off and left its dirt upon it, and it was soon cover’d with mire, which was not remov’d, the city as yet having no scavengers.
Franklin may have learned this usage in England, notice fourth definition below. But also I am baffled by etymology from toot meaning "inspection".

scavenger
n.
5ME scavager < Anglo-Fr scawage, inspection < NormFr escauwer, to inspect < Fl scawen or Frank scouwon, to peer at, observe, akin to OE sceawian, SHOW6
1 a person who gathers things that have been discarded by others, as a junkman
2 any animal that eats refuse and decaying organic matter
3 anything that removes impurities, refuse, etc.
4 [Chiefly Brit.] a person employed to clean the streets, collect refuse, etc.

A scavenger, broadly, looks through discarded material seeking items that are of use to him. He is therefore ‘inspecting’ the material in order to select the useful from the useless.

Further info’ on etymology:

Alteration of Middle English scauager, schavager, official charged with street maintenance, from Anglo-Norman scawager, toll collector, from scawage, a tax on the goods of foreign merchants, from Flemish scauwen, to look at, show. Show: Old English scEawian to look, look at, see; akin to Old High German scouwOn to look, look at, and probably to Latin cavEre to be on one's guard.
Date: 12th century.