Hmm... not necessarily. All that is required, at least at my university, is consent from the participants, and additional paperwork involved in sharing audio/video. If all field linguists were to get that permission before recording, then they could share later, as long as consent was involved.
In fact, I think this is the direction the discipline is headed.
Another problem has been technology, where audio recording was expensive or impractical for years, so it was not done all the time, and now even video recording is easy. You can fit 1000 hours easily on a single hard drive, without needing tapes or other storage.
But then the issue is again privacy: do these speakers want their recordings shared? It's very important that we respect what they want. Archaeology doesn't have that problem: no 3,000 year old mummy is going to complain if we post its picture on Wikipedia.
Regardless, some material is in fact available. There are various archives that host it, although sometimes it's in limited amounts.
Another question is how valuable sharing everything would be. Maybe allowing access to everything would help some, but usually it's just other researchers who want that much access. A sample of recordings is usually enough for the general public. And researchers often do share with other researchers-- so if you're working on a project, you could ask someone about their data.