I'm pretty sure you can still pay someone to pick up and dispose of your trash regardless of any public trash disposal service regime.

Like how you can hire private bodyguards, and not leave your safety to the police.

Or course your trash stopping being "picked up reliably" and the presence of alternative private services are often not independent events (either the government/local government decides to not put in the effort and leave it to the public to pay for third party services, or they're actively encouraged to do so with under the table bribes).

You can have "premium" services for-profit over and above base-level services. In countries where the government provides healthcare, you can pay for amenities like a private hospital room. Or if the government is too slow with mail delivery, you can pay a courier to overnight it. and so on.