TIdMessage.Recipients.EMailAddresses被设计为包含多个电子邮件地址的单个分隔字符串，但您在循环中反复为其分配单个电子邮件地址，因此它将只包含您分配的最后一个电子邮件地址。
在这种情况下，请使用TIdMessage.Recipients.Add()而不是TIdMessage.Recipients.EMailAddresses。
此外，您没有正确地枚举TListView项。如果引发异常，您并不是在保护代码不会泄漏内存。
试着这样做：

var
  smtpServer: TIdSMTP;
  msgEmail: TIdMessage;
  ssl: TIdSSLIOHandlerSocketOpenSSL;
  i: integer;
begin
  smtpServer := TIdSMTP.Create(nil);
  try
    msgEmail := TIdMessage.Create(smtpServer);

    msgEmail.Sender.Address := edtSenderAddress.Text;
    msgEmail.Sender.Name := edtSenderName.Text;
    msgEmail.ReplyTo.EMailAddresses := edtReplyTo.Text;

    for i := 0 to lsvRecipients.Count-1 do
    begin
      msgEmail.Recipients.Add.Address := lsvRecipients.Items[i].Caption;
      // or:
      // msgEmail.Recipients.Add.Text := lsvRecipients.Items[i].Caption;
    end;

    msgEmail.Subject := edtSubject.Text;
    msgEmail.Body.Text := redBody.Text;

    ssl := TIdSSLIOHandlerSocketOpenSSL.Create(smtpServer);
    ssl.SSLOptions.Method := sslvTLSv1_2;
    ssl.SSLOptions.Mode := sslmUnassigned;
    ssl.SSLOptions.VerifyMode := [];
    ssl.SSLOptions.VerifyDepth := 0;

    smtpServer.IOHandler := ssl;
    smtpServer.Host := edtHost.Text;
    smtpServer.Port := StrToInt(edtPort.Text);
    smtpServer.Username := edtUsername.Text;
    smtpServer.Password := edtPassword.Text;
    smtpServer.UseTLS := utUseExplicitTLS;

    smtpServer.Connect;
    try
      smtpServer.Send(msgEmail);
    finally
      smtpServer.Disconnect;
    end;
  finally
    smtpServer.Free;
  end;

  // Output
  Beep;
  ShowMessage('Email Sent!');
end;

procedure TfrmEmailBomber.btnAddEmailClick(Sender: TObject);
var
  emailToAdd: TListItem;
begin
  emailToAdd := lsvRecipients.Items.Add;
  emailToAdd.Caption := edtAddEmail.Text;
end;