Years ago I read an article about how slow water had to flow over a grassy area to prevent erosion and cutting away of the soil. I cannot remember where I saw it, and I don't have the time now to do an extensive search for that information. That information IS out there, somewhere. I dont' know if it is in here or not.
https://damsafety.org/dam-owners/open-channel-spillways

i.e. if a 3' wide 6" deep flow of water was eroding the emergency spillway, making it 18' wide and only 1" deep would slow down the speed of the water flow and stop the erosion.

In any case, I would put the thickest geotextile fabric down on the ground before you put any rocks down, and make sure that the geotextile fabric is under the outflow of any drain pipe or culvert so the water won't go under it.

Off the top of my head I don't know what a 12" siphon pipe will equate to flow wise to an open discharge culvert. I know that a siphon will flow more water than a culvert.

Could you put a gate valve in the 12" siphon pipe to allow you to open/close it when required? Just throwing ideas out there. Let me get in touch with Rex and see if he can weigh in on this. He's done a lot of work with siphon systems, more than I have.