I believe standing on one foot is really underrated and VERY helpful! That was one thing we tried to have my mom do (she was in her 90s). I am having my 80+ year old husband practice it as well (plus I do it). If you can do it well and for a prolonged period, try doing it with your eyes closed. That adds an extra dimension of challenge.
It's important to remember that when we walk, we are basically standing on one foot for a while, so as long as we can balance on one leg, we can walk more normally for longer rather than shuffling along. The podiatrist, OT, PT and everyone says standing on one leg (and alternating) is an excellent exercise and takes basically no extra equipment.
I've fallen stepping out our front door onto a slipper on the doormat and broke a bone. Ouch! I also fell when I was carrying my ipad and walking instead of just focusing on one thing. Caught the fall with my wrist and broke a finger. Boo! Those were the only two bones I broke and it was awhile ago, both post-menopause. I did have a dexascan and both revealed I have mild oseopenia but NOT osteoperosis.
My folks were lucky never to break any bones due to falls ever. They both lived to their 90s, so I still have a ways to go. My older sister has osteoperosis and is taking medication for it--she's fractured ribs and has a very delicate frame. No one else in our family has osteoperosis.
We have multiple loose throw rugs around the house but no one has slipped or tripped with them. They stay put where they are laid. We COULD remove them but we like them as they are.
I could not agree more with you about the practice of balancing on one leg at a time. Suck your gut in and focus on one spot on the wall to help get to longer times fully balanced on one leg. Also walk a bit on your tiptoes and a bit on your heels every day. This does not have to be a time consuming, organized exercise session, but it does have to be done regularly. A little bit is a lot here. Not breaking bones is the reward.