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

routers/web/user/notification.go

@@ -15,6 +15,7 @@ import (
 	"code.gitea.io/gitea/modules/base"
 	"code.gitea.io/gitea/modules/context"
 	"code.gitea.io/gitea/modules/setting"
+	api "code.gitea.io/gitea/modules/structs"
 )
 
 const (
@@ -191,3 +192,8 @@ func NotificationPurgePost(c *context.Context) {
 
 	c.Redirect(setting.AppSubURL+"/notifications", http.StatusSeeOther)
 }
+
+// NewAvailable returns the notification counts
+func NewAvailable(ctx *context.APIContext) {
+	ctx.JSON(http.StatusOK, api.NotificationCount{New: models.CountUnread(ctx.Doer)})
+}