Describe the bug

After #9934 was merged, @williammartin called out a problem where the logic handling extension updates is blocking, which will cause the user to wait an indeterminate amount of time for gh to check if a new version is available:

This was a mistake on the author's part as the update checking logic should be non-blocking like it is with core gh update checking:

Steps to reproduce the behavior

1. Create simple extension for testing

   gh ext create gh-sleep
   cd gh-sleep
   gh repo create --push --private --source .
   gh ext install andyfeller/gh-sleep

2. Artificially extend extension update check behavior

   diff --git a/pkg/cmd/root/extension.go b/pkg/cmd/root/extension.go
   index 7f2325e1..95d4669d 100644
   --- a/pkg/cmd/root/extension.go
   +++ b/pkg/cmd/root/extension.go
   @@ -32,6 +32,9 @@ func NewCmdExtension(io *iostreams.IOStreams, em extensions.ExtensionManager, ex
                   // PreRun handles looking up whether extension has a latest version only when the command is ran.
                   PreRun: func(c *cobra.Command, args []string) {
                           go func() {
   +                               fmt.Fprintf(io.ErrOut, "Artifically delaying update check logic, sleeping for 3 minutes")
   +                               time.Sleep(3 * time.Minute)
   +                               fmt.Fprintf(io.ErrOut, "Artifically delay up!")
                                   releaseInfo, err := checkExtensionReleaseInfo(em, ext)
                                   if err != nil && hasDebug {
                                           fmt.Fprintf(io.ErrOut, "warning: checking for update failed: %v", err)
   

3. Build and run extension, confirming blocking behavior

   make
   time ./bin/gh sleep

   resulting in:

   Artifically delaying update check logic, sleeping for 3 minutes
   Hello gh-sleep!
   Artifically delay up!
   ./bin/gh sleep  0.05s user 0.02s system 0% cpu 3:00.06 total
   

Expected vs actual behavior

Again, extension update checking logic should be non-blocking.

Logs

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