@meghan13 I just watched the video, and drew a slightly different conclusion.

Unlike asbestos, fiberglass and rock wool insulation have been determined to be not permanently harmful because the macrophages in the lungs are able to break them down and carry them away within about 10 days.
Yes, people with sensitive lungs can be bothered and they should limit contact.
Yes, constant and concentrated exposure can be problematic. Fiberglass workers wear respirators and goggles, long sleeves and gloves to protect from overexposure.

The safety agencies have determined that undisturbed fiberglass insulation is safe, but I understand your concern with the uncovered older insulation in your home.
The photo of your insulation looks to me like rock wool, another product altogether. That is or can be irritating to skin, but is safer than fiberglass.
In either case, you can staple builders plastic or other sheathing to the floor joists to cover the exposed insulation. This would be much easier than and safer than replacement. Just be sure whoever does it wears protection, and disturbs the insulation as little as possible. And vacuum thoroughly afterward, while masked, to get rid of any resulting dust.

In the list of risks, unless you spend a lot of time in the basement while people are upstairs and shaking insulation dust loose, I would consider you low on the list.
Sue