Should whole house humidifier leak cold air in the summer?

I’ve recently been learning a lot about sealing our HVAC ducts to improve system efficiency. On my mission to seal every accessible leak, I noticed my air conditioning/furnace cabinet joints were leaking quite a bit of air. I started sealing them all up, but noticed my attached humidifier was one of the biggest culprits of air leakage.

The operating manual says it draws in ambient air through a humidifier pad via a fan into the duct to distribute throughout the house (picture from manual attached). That makes sense, but it seems the same air inlet used during winter operation lets air out when trying to air condition the house. The manual doesn’t mention any sort of "summer operating mode".

So my questions are:

1. Is there a way to prevent air leakage with this kind of unit (Honeywell HE300 if model-specific)?
2. Am I going to a pointless extreme by trying to seal every last cabinet leak?