Never use /api/v1 from Gitea UI Pages (#19318)

Reusing `/api/v1` from Gitea UI Pages have pros and cons.
Pros:
1) Less code copy

Cons:
1) API/v1 have to support shared session with page requests.
2) You need to consider for each other when you want to change something about api/v1 or page.

This PR moves all dependencies to API/v1 from UI Pages.

Partially replace #16052

web_src/js/features/stopwatch.js

@@ -111,7 +111,7 @@ async function updateStopwatchWithCallback(callback, timeout) {
 async function updateStopwatch() {
   const data = await $.ajax({
     type: 'GET',
-    url: `${appSubUrl}/api/v1/user/stopwatches`,
+    url: `${appSubUrl}/user/stopwatches`,
     headers: {'X-Csrf-Token': csrfToken},
   });