Unfortunately, there is only one reason to bypass the heater core... It probably developed a leak.
Post a better picture of the Tee that you're talking about and we can try to identify the heater hoses so that you can connect up the heater core and see if it leaks or not.
In the meantime, as long as your A/C works, the defrost will still function to a certain extent. It's just not anywhere near as efficient as when it has the heat to help it.