tap: actively check repo visibility

We shouldn't assume that the visibility of a tap keeps unchanged by
storing it into `TapConfig`. Instead, let's actively check that to avoid
e.g. collecting unwanted analytics.

Also, speed up the check for known official public taps so that we don't
waste our GitHub API token limit in CI.

Library/Homebrew/tap.rb

@@ -366,23 +366,16 @@ class Tap
     return @private unless @private.nil?
 
     @private = T.let(
-      if (value = config[:private]).nil?
+      begin
-        config[:private] = begin
+        if core_tap? || core_cask_tap? || OFFICIAL_CMD_TAPS.include?(name)
-          if custom_remote?
-            true
-          else
-            # Don't store config if we don't know for sure.
-            return false if (value = GitHub.private_repo?(full_name)).nil?
-
-            value
-          end
-        rescue GitHub::API::HTTPNotFoundError
-          true
-        rescue GitHub::API::Error
           false
+        elsif custom_remote? || (value = GitHub.private_repo?(full_name)).nil?
+          true
+        else
+          value
         end
-      else
+      rescue GitHub::API::Error
-        value
+        true
       end,
       T.nilable(T::Boolean),
     )