That's a great question that cities worldwide grapple with. The short answer is: they can be highly effective, but their success depends entirely on design, implementation, and location. A poorly chosen or installed locking lid is often just a broken lid waiting to happen.
The primary goal of a locking lid is to contain waste—preventing litter from being scattered by wind or animals, and controlling odors. In this, a robust, functional locking mechanism is very effective. It creates a significant barrier for pests like raccoons and rats, a major urban concern. However, the "broken" part of your question points to the human factor.
Vandalism is the chief enemy. A flimsy padlock hasp or a plastic latch can be snapped off easily, rendering the lock useless and leaving a damaged bin. The most successful systems use internal, tamper-resistant mechanisms—like a spring-loaded bar or a internal cam lock—that have no external parts to pry or break. Heavy-duty materials like thick gauge steel or reinforced polymer are non-negotiable for high-traffic areas.
Placement is critical. In a monitored park or a corporate district, locking lids thrive. In a neglected alley, they may simply present a challenge for vandals. Maintenance is also key; a jammed lock that people force open will break.
So, are they effective? Yes, when they are part of a holistic system: a well-built bin, an intelligent locking design suited to the threat level, and consistent maintenance. A cheap lock on a cheap bin is largely a placebo. A properly engineered one is a vital tool for cleaner, more sanitary, and more efficient urban waste management. The breakage happens not because the concept is flawed, but because of execution. Cities must invest in quality over a false sense of security.