Any footbridge or small vehicle bridge that I have seen around here has eventually washed out by a larger than expected rainfall. Even one the electric co-op built for a public park with big power poles to hold the bridge. Our creek bed is nearly the same size as the OP's, just a little larger, maybe 12 feet wide. With a 500 acre watershed, any bridge would be a major project, like public works would do for a road. I can cross with my walk behind brush mower by putting up a ramp on the steep side and going down a shallow slope on the other. No tractor access unless through the neighbor's property. I only have about a half acre on the other side of the creek, no it doesn't matter much, as it's all low lying.
It depends on the watershed the OP's creek has and the largest rain expected. Our 12 foot wide creek bed spills over and becomes probably 40 feet wide every two or three years.
I would say the OP has a good chance of a washout with that small bridge, unless the bed is very very deep under the bridge. It's not so much the water that destroys things. It's the debris. Our creek can carry 30 foot trees downstream and has recently rolled 300 pound rocks.