What you are describing appears to be the use of user stories as requirements in the ISO/IEC/IEEE 29148 sense.
Although it's not uncommon for people to consider the user story to be the "as a {role}, I want {goal} so that {objective}", that's just the card part of a user story. A user story is also a conversation between the team and stakeholders, which results in an understanding of what is necessary or expected. The conversation between the team and the stakeholders results in things like acceptance criteria, mockups and wireframes, and other notes that help guide the development. Finally, the user story includes confirmation that the objectives have been met. These are the Three Cs of User Stories.
The evolution of a user story from a card through the conversation through the confirmation mirrors the lifecycle of requirements from discovery through elicitation and then the use in design, construction, verification, and validation of the system described in standards such as ISO/IEC/IEEE 29148 and other more traditional descriptions of requirements engineering.
However, requirements management is also a part of requirements engineering. Requirements management is about making sure that each requirement, as well as the overall set of requirements, maintains the necessary state over time. In particular, individual requirements and the set of requirements are expected to be complete and consistent at every point in time.
The idea of updating user stories, their associated acceptance criteria, and any relevant test cases is a way to implement requirements management of user stories. If you have software source code (commits, pull requests) and test cases also linked to the user stories and acceptance criteria, you also satisfy the traceability aspects of requirements engineering.
In theory, there's nothing wrong with this approach. In some cases, it may even be desired or necessary to have a robust requirements engineering process. In reality, the problems that you present concerning having to search for the related user story (or stories) for updating can be painful. Some tools may make this easier, but it would require applying consistent metadata to issues.
For some systems, this approach may work fine. I'm looking primarily at bespoke software systems that are built for one specific client against a specification with a limited number of user roles where you'd have an easier time deconflicting requirements. In other cases, such as developing commercial software for the market where you are building to needs rather than to spec, it's much harder to manage. Many of the requirements engineering techniques that I've come across are better suited for building to spec rather than building for market needs.
When building systems for need, I tend to focus on the ability to generate an as-built specification rather than a to-build specification. Instead of specifying the requirements that must be satisfied, specify the behavior (functional) and quality attributes (non-functional) characteristics of the system. Tools that allow you to write BDD-style test cases offer a good way to generate these, especially for test cases at the system level. Managing the tests becomes more important than managing the requirements.