It appears that the third-party API is not receiving any requests from either the web or mobile interfaces

I have integrated my own server’s login interface as a third-party API, and I have also set up the business flow and endpoints for it. I have enabled it in both the web and mobile applications. However, after publishing the app, when users enter their account credentials and click the login button on the web interface, it does not send a request to my server. The same issue occurs on the mobile app, where clicking the login button does not trigger any response.

First of all need to check that business process really starts when you send request to endpoint. Probably you should add Log block before API request and check it in Deploy Stats → Application logs.
Another way is using HTTP Request block insted of External API Request from designer. May be it will be more usable in your case. Logging Success and Error of request also should help to understand the issue.

Before calling the API within the business process, I attempted to set up logging. I also added logging statements at the beginning of the button’s click event. However, when I ran the code, I couldn’t find the logs that I expected to be printed. I’m unsure about the root cause of this issue.

Currently your log block is not used. It doesn’t have In flow connected and never starts

I made changes according to your instruction, but I still can’t find the log I recorded, and it is still reporting the same JavaScript error.jobRunner-l5kHXSJW.js:57 ReferenceError: structuredClone is not defined
at pV (jobRunner-l5kHXSJW.js:57:14924)
at onmessage (jobRunner-l5kHXSJW.js:57:15711)