Note that brew update will only report all without API.

It's not worth supporting this option any longer given the API
constraints.

Library/Homebrew/cmd/update-report.rb

@@ -629,7 +629,12 @@ class ReporterHub
   delegate empty?: :@hash
 
   def dump(updated_formula_report: true)
-    report_all = Homebrew::EnvConfig.update_report_all_formulae?
+    report_all = ENV["HOMEBREW_UPDATE_REPORT_ALL_FORMULAE"].present?
+    if report_all && !Homebrew::EnvConfig.no_install_from_api?
+      odeprecated "HOMEBREW_UPDATE_REPORT_ALL_FORMULAE"
+      opoo "This will not report all formulae because Homebrew cannot get this data from the API."
+      report_all = false
+    end
 
     dump_new_formula_report
     dump_new_cask_report

Library/Homebrew/env_config.rb

@@ -378,11 +378,6 @@ module Homebrew
         default_text: "macOS: `/private/tmp`, Linux: `/tmp`.",
         default:      HOMEBREW_DEFAULT_TEMP,
       },
-      HOMEBREW_UPDATE_REPORT_ALL_FORMULAE:       {
-        description: "If set, `brew update` lists changes to all formulae and cask files rather than only showing " \
-                     "when they are new and not installed or outdated and installed.",
-        boolean:     true,
-      },
       HOMEBREW_UPDATE_TO_TAG:                    {
         description: "If set, always use the latest stable tag (even if developer commands " \
                      "have been run).",

Library/Homebrew/sorbet/rbi/hidden-definitions/hidden.rbi

@@ -4415,8 +4415,6 @@ module Homebrew::EnvConfig
 
   def self.temp(); end
 
-  def self.update_report_all_formulae?(); end
-
   def self.update_to_tag?(); end
 
   def self.verbose?(); end

docs/Manpage.md

@@ -2268,9 +2268,6 @@ example, run `export HOMEBREW_NO_INSECURE_REDIRECT=1` rather than just
 
   *Default:* macOS: `/private/tmp`, Linux: `/tmp`.
 
-- `HOMEBREW_UPDATE_REPORT_ALL_FORMULAE`
-  <br>If set, `brew update` lists changes to all formulae and cask files rather than only showing when they are new and not installed or outdated and installed.
-
 - `HOMEBREW_UPDATE_TO_TAG`
   <br>If set, always use the latest stable tag (even if developer commands have been run).
 

manpages/brew.1

@@ -3357,12 +3357,6 @@ Use this path as the temporary directory for building packages\. Changing this m
 \fIDefault:\fR macOS: \fB/private/tmp\fR, Linux: \fB/tmp\fR\.
 .
 .TP
-\fBHOMEBREW_UPDATE_REPORT_ALL_FORMULAE\fR
-.
-.br
-If set, \fBbrew update\fR lists changes to all formulae and cask files rather than only showing when they are new and not installed or outdated and installed\.
-.
-.TP
 \fBHOMEBREW_UPDATE_TO_TAG\fR
 .
 .br