Does "Connect" statement for every query indicate broken connection pooling?
Three aspects must be considered
ASPECT #1 : PHP
PHP has a setting for Persistent Connections
[MySQL]
; Allow or prevent persistent links.
mysql.allow_persistent=On

Make sure this is on in php.ini if you want to use Persistent Connections
ASPECT #2 : Your Code
Since mysql_pconnect is deprecated, make sure you are using one of the following

mysqli-connect with p: before the host name
PDO::__construct() with PDO::ATTR_PERSISTENT as a driver option

ASPECT #3 : MySQL
When mysqld sees a connection attempt, it records it statistically as one of the following

Connections : Connections Attempts (Successful or Not)
Aborted_connects : Failed Connection Attempts

If the general log is echoing just the word Connect as an event, chances are that's just a Connection attempt. Any subsequent events on that connection would be operating on a established thread (DB Connection). The general log will not tell you whether the DB Connection is newly created or already established.
To actually know if the connections are coming and going, you should monitor

Threads_connected : The number of currently open connections
Threads_created : The number of threads created to handle connections

To actually see them in a single query, please run this
SELECT * FROM information_schema.global_status WHERE variable_name IN
('Connections','Aborted_connects','Threads_connected','Threads_created');