I think the free service was born out of despair for people consistently leaving big trash on the roadside. While I understand you as I don't produce big trash myself, I guess it's still better to just pay a little more and get rid of the bigger problems having trash on the roadside would bring.
New York City fines people $100 to $400 for using improper receptacles, leaving receptacles uncovered, putting trash in front of others' buildings and/or leaving loose rubbish lying about [1]. Note that these can accumulate, e.g. if you put loose trash in front of someone else's building in an improper receptacle, that could be $300 to $1,000 of fines.
Those fines pay for enforcement. They also pay for picking up garbage dumped by unknown persons. This model shifts the burden from everyone to just the offenders. I find it preferable to small households subsidising big ones.
How do you demonstrate that the fridge left on the side of a public road, even if close to someone's home, actually belonged to someone? The examples that you mention are even harder to enforce. Cameras?
My house has never had a fire, so why should I pay for a fire department in my neighborhood?
That's what your argument basically boils down to.
Obviously it's not in our collective best interest to have trash accumulate on the streets. So we all end up paying for it collectively, regardless of how much we consume the service as individuals.
I have less control over whether my house lights on fire than if I make lots of trash. Also, if my house starts burning, it's an immediate threat to my neighbours' safety. If I leave trash outside, it's a less-immediate threat to their comfort.
Economically, clean streets are non-rivalrous (my clean street costs little more than my neighbor's) and retroactively excludable (through fines). This makes it a public enterprise good, akin to mail or trains [1].
Case in point: mattress and A/C sellers in New York City dispose of your old mattress or A/C for you. If pick-up of such things were free, fewer would do that. One response might be to regulate every thing one might purchase that could produce big trash. The other is to just make people who choose to make big trash pay for it.
I don't produce lots of big trash. Not sure why I--nor anybody else--should subsidise people who produce lots of big trash.