In my work, I do requirements gathering, analysis and design of business solutions in addition to coding. There are multiple software systems and packages, and developers are expected to work on any of them, instead of being assigned to make changes to only 1 system or just a few systems.

How developers ensure they have captured all of the necessary requirements and resolved any conflicting requirements?

An example of this type of scenario:

Bob the developer is asked to modify the problem ticket system for a hypothetical utility repair business. They contract with a local utility company to provide this service. The old system provides a mechanism for an external customer to create a ticket indicating a problem with utility service at a particular address. There is a scheduling system and an invoicing system that is dependent on this data. Bob's new project is to modify the ticket placement system to allow for multiple addresses to entered by a landlord or other end customer with multiple properties. The invoicing system bills per ticket, but should be modified to bill per address. What practices would help Bob discover that the invoicing system needs to be changed as well? How might Bob discover what other systems in his company might need to be changed in order to support the new changes\business model? Let's say there is a documented specification for each system involved, but there are many systems and Bob is not familiar with all of them.

End of example.

We're often in this scenario, and we do have design reviews but management places ultimate responsibility for any defects (business process or software process) on the developer who is doing the design and the work.

Some organizations seem to be better at this than others. How do they manage to detect and solve conflicting or incomplete requirements across software systems?

We currently have a lot of tribal knowledge and just a few developers who understand the entire business and software chain. This seems highly ineffective and leads to problems at the requirements level.