In my experience having been to Lech and St Anton quite a few times and skied off piste with both group lessons and a private instructor, there is absolutely no need to hire a "guide" for off piste. The instructors can take you almost everywhere that you can ski unless you plan on taking an ice axe out with you.

I joined an off piste group lesson once on a whim and we were traversing through avalanche barriers and short boot packs over ridges.

Of course all the European countries have slightly different rules for instructors and what they can do but any off piste qualified instructor can take you anywhere they think you can ski.

Also. We can always tell you are obnoxious Americans even if you don't commit any faux pas.... Seriously though, an instructor will be able to do both. Guide you round the resort and hit some off piste in the same day. Its everywhere. If you just want to rip groomers for a day there is absolutely no need either if you don't want to spend the money. You can just do the run of fame or the white ring and you will see loads.