The way you do every day tasks with your horse matters
How do you get your horse responding softly in a snaffle? By asking for softness in every interaction, be it with your hand when you touch, or lead rope and halter when you handle.
A horse carries the feel from one to the other. If they drag on your lead rope, if you push and pull on them, then it only makes sense to them to push and pull on your hands, resist your legs, and brace against you.
There are so many opportunities to work on getting your horse to soften and feel good about your interactions together. When I feed, I ask my horses to step back and soften before I put their feed bags on, as if I were bridling.
When I halter, I ask them to lower their heads and tip their noses toward me.
When I lead, I ask them to leave slack in the rope and follow the feel of the rope, to engage in a conversation with me, instead of wallowing around behind me.
In this way, by the time I swing a leg over, their minds are already geared toward a soft response, and a meaningful engagement.