Only create LazyContentProvider for the non-rendering Site

Which saves a fair amound of allocations:

```
gobench --package ./hugolib --bench "SiteNew/Regular_D" --base master
```

Before:

```
name                                  old time/op    new time/op    delta
SiteNew/Regular_Deep_content_tree-10    40.7ms ± 3%    41.2ms ± 1%    ~     (p=0.343 n=4+4)

name                                  old alloc/op   new alloc/op   delta
SiteNew/Regular_Deep_content_tree-10    27.7MB ± 0%    28.8MB ± 0%  +3.76%  (p=0.029 n=4+4)

name                                  old allocs/op  new allocs/op  delta
SiteNew/Regular_Deep_content_tree-10      304k ± 0%      329k ± 0%  +8.07%  (p=0.029 n=4+4)
```

After:

```
name                                  old time/op    new time/op    delta
SiteNew/Regular_Deep_content_tree-10    34.2ms ± 1%    34.7ms ± 1%    ~     (p=0.114 n=4+4)

name                                  old alloc/op   new alloc/op   delta
SiteNew/Regular_Deep_content_tree-10    27.7MB ± 0%    28.1MB ± 0%  +1.38%  (p=0.029 n=4+4)

name                                  old allocs/op  new allocs/op  delta
SiteNew/Regular_Deep_content_tree-10      304k ± 0%      314k ± 0%  +3.03%  (p=0.029 n=4+4)
```

Updates #8919

hugolib/page.go

@@ -940,20 +940,6 @@ func (p *pageState) shiftToOutputFormat(isRenderingSite bool, idx int) error {
 		panic(fmt.Sprintf("pageOutput is nil for output idx %d", idx))
 	}
 
-	// We attempt to assign pageContentOutputs while preparing each site
-	// for rendering and before rendering each site. This lets us share
-	// content between page outputs to conserve resources. But if a template
-	// unexpectedly calls a method of a ContentProvider that is not yet
-	// initialized, we assign a LazyContentProvider that performs the
-	// initialization just in time.
-	p.pageOutput.ContentProvider = page.NewLazyContentProvider(func() (page.ContentProvider, error) {
-		cp, err := newPageContentOutput(p, p.pageOutput)
-		if err != nil {
-			return nil, err
-		}
-		return cp, nil
-	})
-
 	// Reset any built paginator. This will trigger when re-rendering pages in
 	// server mode.
 	if isRenderingSite && p.pageOutput.paginator != nil && p.pageOutput.paginator.current != nil {
@@ -985,7 +971,24 @@ func (p *pageState) shiftToOutputFormat(isRenderingSite bool, idx int) error {
 			}
 		}
 		p.pageOutput.initContentProvider(cp)
-		p.pageOutput.cp = cp
+	} else {
+		// We attempt to assign pageContentOutputs while preparing each site
+		// for rendering and before rendering each site. This lets us share
+		// content between page outputs to conserve resources. But if a template
+		// unexpectedly calls a method of a ContentProvider that is not yet
+		// initialized, we assign a LazyContentProvider that performs the
+		// initialization just in time.
+		if lcp, ok := (p.pageOutput.ContentProvider.(*page.LazyContentProvider)); ok {
+			lcp.Reset()
+		} else {
+			p.pageOutput.ContentProvider = page.NewLazyContentProvider(func() (page.ContentProvider, error) {
+				cp, err := newPageContentOutput(p, p.pageOutput)
+				if err != nil {
+					return nil, err
+				}
+				return cp, nil
+			})
+		}
 	}
 
 	return nil

resources/page/page_lazy_contentprovider.go

@@ -49,6 +49,10 @@ func NewLazyContentProvider(f func() (ContentProvider, error)) *LazyContentProvi
 	return &lcp
 }
 
+func (lcp *LazyContentProvider) Reset() {
+	lcp.init.Reset()
+}
+
 func (lcp *LazyContentProvider) Content() (interface{}, error) {
 	lcp.init.Do()
 	return lcp.cp.Content()
@@ -67,35 +71,29 @@ func (lcp *LazyContentProvider) PlainWords() []string {
 func (lcp *LazyContentProvider) Summary() template.HTML {
 	lcp.init.Do()
 	return lcp.cp.Summary()
-
 }
 
 func (lcp *LazyContentProvider) Truncated() bool {
 	lcp.init.Do()
 	return lcp.cp.Truncated()
-
 }
 
 func (lcp *LazyContentProvider) FuzzyWordCount() int {
 	lcp.init.Do()
 	return lcp.cp.FuzzyWordCount()
-
 }
 
 func (lcp *LazyContentProvider) WordCount() int {
 	lcp.init.Do()
 	return lcp.cp.WordCount()
-
 }
 
 func (lcp *LazyContentProvider) ReadingTime() int {
 	lcp.init.Do()
 	return lcp.cp.ReadingTime()
-
 }
 
 func (lcp *LazyContentProvider) Len() int {
 	lcp.init.Do()
 	return lcp.cp.Len()
-
 }