how to query SSISDB to find out the errors in the packages?
I have a handful of queries I use. The general concepts are that the table with the information is catalog.operation_messages and you are interested in events with a 120 (error) type. 

Depending on how robust of a query you want to build out, the following two derived tables might also be of interest. 

--- http://technet.microsoft.com/en-us/library/ff877994.aspx
-- This query translates the message_type from SSISDB.catalog.operation_messages
-- into useful text
SELECT
    D.message_type
,   D.message_desc
FROM
(
    VALUES
        (-1,'Unknown')
    ,   (120,'Error')
    ,   (110,'Warning')
    ,   (70,'Information')
    ,   (10,'Pre-validate')
    ,   (20,'Post-validate')
    ,   (30,'Pre-execute')
    ,   (40,'Post-execute')
    ,   (60,'Progress')
    ,   (50,'StatusChange')
    ,   (100,'QueryCancel')
    ,   (130,'TaskFailed')
    ,   (90,'Diagnostic')
    ,   (200,'Custom')
    ,   (140,'DiagnosticEx Whenever an Execute Package task executes a child package, it logs this event. The event message consists of the parameter values passed to child packages.  The value of the message column for DiagnosticEx is XML text.')
    ,   (400,'NonDiagnostic')
    ,   (80,'VariableValueChanged')
) D (message_type, message_desc);


-- Where was the error message generated?
SELECT
    D.message_source_type
,   D.message_source_desc
FROM
(
    VALUES
        (10,'Entry APIs, such as T-SQL and CLR Stored procedures')
    ,   (20,'External process used to run package (ISServerExec.exe)')
    ,   (30,'Package-level objects')
    ,   (40,'Control Flow tasks')
    ,   (50,'Control Flow containers')
    ,   (60,'Data Flow task')
) D (message_source_type, message_source_desc);


I use queries like this to find information about the errors. Maybe I only care about what the errors were (query 1). Other times, I want to know all the activities of all the operations that failed (query 2). Generally, I'm lazy and want to see all the information about the last failing operation (query 3 and note the caveat).

-- http://msdn.microsoft.com/en-us/library/ff877994.aspx
-- Find all error messages
SELECT
    OM.operation_message_id
,   OM.operation_id
,   OM.message_time
,   OM.message_type
,   OM.message_source_type
,   OM.message
,   OM.extended_info_id
FROM
    catalog.operation_messages AS OM
WHERE
    OM.message_type = 120;

-- Generate all the messages associated to failing operations
SELECT
    OM.operation_message_id
,   OM.operation_id
,   OM.message_time
,   OM.message_type
,   OM.message_source_type
,   OM.message
,   OM.extended_info_id
FROM
    catalog.operation_messages AS OM
    INNER JOIN
    (  
        -- Find failing operations
        SELECT DISTINCT
            OM.operation_id  
        FROM
            catalog.operation_messages AS OM
        WHERE
            OM.message_type = 120
    ) D
    ON D.operation_id = OM.operation_id;

-- Find all messages associated to the last failing run
SELECT
    OM.operation_message_id
,   OM.operation_id
,   OM.message_time
,   OM.message_type
,   OM.message_source_type
,   OM.message
,   OM.extended_info_id
FROM
    catalog.operation_messages AS OM
WHERE
    OM.operation_id = 
    (  
        -- Find the last failing operation
        -- lazy assumption that biggest operation
        -- id is last. Could be incorrect if a long
        -- running process fails after a quick process
        -- has also failed
        SELECT 
            MAX(OM.operation_id)
        FROM
            catalog.operation_messages AS OM
        WHERE
            OM.message_type = 120
    );


Perhaps I'm lazy and don't want to go look this information up in the event of a failure, much as your team appears to have done. I have a SQL Agent job that runs on demand and I have jobs that run SSIS packages set to run that job in the event of failure.

DECLARE
    @profile_name sysname = 'SQLAdmins'
,   @recipients varchar(max) = 'billinkc@kfc.com'
,   @copy_recipients varchar(max) = NULL
,   @blind_copy_recipients varchar(max) = NULL
,   @subject nvarchar(255) = 'failed package test'
,   @body nvarchar(max) = 'Stuff has failed, fix please'
,   @body_format varchar(20) = NULL
,   @importance varchar(6) = 'NORMAL'
,   @sensitivity varchar(12) = 'NORMAL'
,   @file_attachments nvarchar(max) = NULL
,   @query nvarchar(max) = N'
SELECT
    O.object_name AS FailingPackageName
,   O.object_id
,   O.caller_name
,   O.server_name
,   O.operation_id
,   OM.message_time
,   EM.message_desc
,   D.message_source_desc
,   OM.message
FROM
    SSISDB.catalog.operation_messages AS OM
    INNER JOIN
        SSISDB.catalog.operations AS O
        ON O.operation_id = OM.operation_id
    INNER JOIN
    (
        VALUES
            (-1,''Unknown'')
        ,   (120,''Error'')
        ,   (110,''Warning'')
        ,   (70,''Information'')
        ,   (10,''Pre-validate'')
        ,   (20,''Post-validate'')
        ,   (30,''Pre-execute'')
        ,   (40,''Post-execute'')
        ,   (60,''Progress'')
        ,   (50,''StatusChange'')
        ,   (100,''QueryCancel'')
        ,   (130,''TaskFailed'')
        ,   (90,''Diagnostic'')
        ,   (200,''Custom'')
        ,   (140,''DiagnosticEx Whenever an Execute Package task executes a child package, it logs this event. The event message consists of the parameter values passed to child packages.  The value of the message column for DiagnosticEx is XML text.'')
        ,   (400,''NonDiagnostic'')
        ,   (80,''VariableValueChanged'')
    ) EM (message_type, message_desc)
        ON EM.message_type = OM.message_type
    INNER JOIN
    (
        VALUES
            (10,''Entry APIs, such as T-SQL and CLR Stored procedures'')
        ,   (20,''External process used to run package (ISServerExec.exe)'')
        ,   (30,''Package-level objects'')
        ,   (40,''Control Flow tasks'')
        ,   (50,''Control Flow containers'')
        ,   (60,''Data Flow task'')
    ) D (message_source_type, message_source_desc)
        ON D.message_source_type = OM.message_source_type
WHERE
    OM.operation_id = 
    (  
        SELECT 
            MAX(OM.operation_id)
        FROM
            SSISDB.catalog.operation_messages AS OM
        WHERE
            OM.message_type = 120
    )
    AND OM.message_type IN (120, 130);
'
,   @execute_query_database sysname = NULL
,   @attach_query_result_as_file bit = 0
,   @query_attachment_filename nvarchar(260) = NULL
,   @query_result_header bit = 1
,   @query_result_width int = 256
,   @query_result_separator char(1) = char(13)
,   @exclude_query_output bit  = 0
,   @append_query_error bit = 0
,   @query_no_truncate bit = 0
,   @query_result_no_padding bit = 0
,   @mailitem_id int = NULL
,   @from_address varchar(max) = NULL
,   @reply_to varchar(max) = NULL;

-- Send email about the failure    
EXECUTE msdb.dbo.sp_send_dbmail
    @profile_name 
,   @recipients
,   @copy_recipients
,   @blind_copy_recipients
,   @subject
,   @body
,   @body_format
,   @importance
,   @sensitivity
,   @file_attachments
,   @query
,   @execute_query_database
,   @attach_query_result_as_file
,   @query_attachment_filename
,   @query_result_header
,   @query_result_width
,   @query_result_separator
,   @exclude_query_output
,   @append_query_error
,   @query_no_truncate
,   @query_result_no_padding
,   @mailitem_id OUTPUT
,   @from_address
,   @reply_to;


Adjust as you like