I only raised chicks in a brooder once and swore never again. Since then I only use broody hens.
If you don't like messing with an indoor brooder then go out and purchase a reliable broody (show breeders or local chicken breeders with lots of birds usually have some they will sell). She will do...
IMO the three week olds are way too young to go out at all, but most especially in a pen with older pullets. People typically don't put them out until they are fully feathered at 5-6 weeks and mixing with older birds is a whole other issue.
I am sure they are getting somewhat messy but that is...