Refactor web package and context package (#25298)

1. The "web" package shouldn't depends on "modules/context" package,
instead, let each "web context" register themselves to the "web"
package.
2. The old Init/Free doesn't make sense, so simplify it
* The ctx in "Init(ctx)" is never used, and shouldn't be used that way
* The "Free" is never called and shouldn't be called because the SSPI
instance is shared

---------

Co-authored-by: Giteabot <teabot@gitea.io>

modules/context/private.go

@@ -11,6 +11,8 @@ import (
 
 	"code.gitea.io/gitea/modules/graceful"
 	"code.gitea.io/gitea/modules/process"
+	"code.gitea.io/gitea/modules/web"
+	web_types "code.gitea.io/gitea/modules/web/types"
 )
 
 // PrivateContext represents a context for private routes
@@ -21,6 +23,12 @@ type PrivateContext struct {
 	Repo *Repository
 }
 
+func init() {
+	web.RegisterResponseStatusProvider[*PrivateContext](func(req *http.Request) web_types.ResponseStatusProvider {
+		return req.Context().Value(privateContextKey).(*PrivateContext)
+	})
+}
+
 // Deadline is part of the interface for context.Context and we pass this to the request context
 func (ctx *PrivateContext) Deadline() (deadline time.Time, ok bool) {
 	if ctx.Override != nil {