My grandfather, with whom I was very close, suffered from Parkinson's in his last decade or so. For a long time he was doing OK: Occasional confusion and the slow, shuffling walk that is characteristic of the disease.
One day he had a minor operation that left him needing a wheelchair for what we thought would be just a few weeks. But he never regained his strength and was never to walk again, which led to a steep and sudden decline in his mental condition. It was truly devastating to see one of the sharpest people I knew become an angry and confused simulacrum of the man I so admired.
I wish I had realized two things then: First, as you say, maintaining mobility is the crucial to the well-being of the elderly. Second, immediate physical/occupational therapy after a fall or surgery is essential to people at risk of losing mobility. Sadly it wasn't offered to us and we didn't think to ask.