I have no problem with the open tee. that's standard procedure for condensate drains, as long as the open tee is DOWNSTREAM of the trap. it vents the drain, and gives a place for cleaning out the trap.
The trap "especially on a negative pressure AH" depends on the siphon effect to drain, you lose that "some" of that effect with the open T. If you close the T and it will draw/siphon the water all the way to the main drain, you end up having more volume of water drawing the water through the trap. Personally, I never leave any T open for critters to crawl into, a small mouse could get into a 3/4" opening, maybe even a large mouse could.