I wouldn't worry in the least about your poultry eating the waste goat feed.
You can always leave them with some limestone/oystershell grit to ensure adequate calcium uptake.
Goat feed will be carefully formulated to ensure no taint that could affect the milk, so the chances of it affecting egg taste is less than remote.
Have you actually seen what free range poultry eat?
Allsorts of animal faeces from dogs to ruminants is pecked at with relish, and free range eggs are renowned for their flavour.
If your hens eat a lot of goat feed it won't do them any harm but may affect egg production as it may be higher in fibre and lower in quality protein than proprietry poultry feed. But bearing in mind your cost saving over throwing the goat feed away, then it probably won't be a problem.
I would however, maybe think about changing the goat feed (as well as the feeder) as there seems little point in buying something that ends up on the floor.
(Just a thought - surely a wider, more shallow trug makes it easier for them to throw the food out? I would use a deeper container.)