Wire gate carabiners used for protection in rock climbing are part of a highly redundant system... protection is placed by the lead climber in a manner to allow minimal drop in the event that one or more fail. A lot of highly dynamic rope in the system ensures shock loading is minimized. Used correctly, this is acceptable for sport climbing... but an arborist doesn't work this way. We're not hanging by our fingertips and one big toe, trying to put a rope through a 'biner. If, starting at your TIP, you placed a canopy anchor every few feet going to many branches and down the trunk all the way to the ground, and half of your SRT line went through them all to a basal tie, then you could probably get by with wiregate carabiners. That's not realistic, of course.
The biggest danger with screwgate carabiners is not contact with the tree opening them.. or running rope going through the 'biner unscrewing them... it's you forgetting to screw the lock barrel tight! The first autolocking carabiners were designed to address this. Since then, double and triple action 'biners were further attempts to increase safety by minimizing the chances of the gate opening due to interaction with objects and with any rope running through the 'biner. Oddly enough, you can open most of them quite easily with a 2' piece of rope if you have someone hold the carabiner (gate facing upward). It seems that running rope outside of the 'biner that comes in contact with the gate presents the biggest chance of opening it. Of course, the 'biner would have to jump out of whatever it was hooked to at the same instant. This would seem to be unlikely, and most accidents aren't from the carabiner failing... they're from careless or improper usage. Climbers who rely solely on the sound of the carabiner or snap on their lanyard clicking into place on their side D rings are really at risk. You really should look at it to make sure that it's actually on there. It's a horribly easy thing to think it went home, only to get that sick feeling a moment later as you fall backwards out of the tree.