it must have seemed really pointless and inconvenient to anyone without a modern concept of hygiene?
Here's a great article from UCLA's dept. of epidemiology regarding contemporary opinions.
http://www.ph.ucla.edu/epi/snow/removal.html
Snow wasn't widely believed at the time of his initial request to shut down the pump, but the governing body, local officials, friends, and physicians agreed to remove the pump handle out of caution. This (famously) [may have, though the outbreak was already in decline] stayed the epidemic.
Unfortunately, my basic research on the matter doesn't report the feelings of the community surrounding the pump, and as many of the poor did not have a voting right in the Broadstreet Guardians (they did not own property), they seem to have been ignored.