In most climates you can actually harvest a modest amount of grass and other outdoor greenery through most of the winter. You need to look somewhere that has tall standing vegetation, like a meadow or fallow field or unmown roadside. Shovel away snow as needed and part the standing dead stuff with your hands and you will discover a surprising amount of relatively live, relatively green stuff down there to be picked. I excavate this for treats for the horses occasionally during the winter, and also for the chickens (although honestly my chickens mostly get produce scraps since they're so much easier to get in quantity
)
Baling grass clippings, you'd have to be real real careful. Either you're producing hay, in which case they need to be quite dry before storing and they need to STAY that way, otherwise you will get mold and respiratory disorders; or you're producing silage, which I have no idea how silage agrees with bird tummies but I would seriously worry about botulism (not an issue for ruminants, but last time I checked, poultry are not ruminants
) Heck, if you want to be feeding dried not fresh grass, just go buy a bale of good hay, that's what it is
Note that some people have had chickens get impacted crops on hay, although others haven't.
I'll stick with mostly produce scraps. Sprouting wheat and such sounds like a really good idea too, although dunno if I'll ever get round to it myself.
Pat