Consolidate fragment reading into _readTemplateFragment

Replace eight bespoke implementations for reading fragments with the existing `_readTemplateFragment()` method.

This enforces the consistent use of the template method design pattern, reduces code duplication, and makes future maintenance easier.

Change-Id: If213b146e4b055523432c6d12f36994cac8f38c1
Reviewed-on: https://dart-review.googlesource.com/c/sdk/+/436820
Reviewed-by: Paul Berry <[email protected]>
Reviewed-by: Konstantin Shcheglov <[email protected]>
Commit-Queue: Paul Berry <[email protected]>

Author: fshcheglov