That is a difficult question to answer. The first thing is to throughly inspect the places where the wood is moving.

It looks as if you may have access to the underside of the stairs, if you do then you will be able to tap the wedges in as they are probably why you have movement. The stairs are cheaply finished as shown by the bandsaw marks still showing on all the treads and risers. While not proof that the original makers skipped steps in production, it certainly makes it a significant possibility. This can be both good and not so great, the good point is that tightening the treads up may be easy, the not great point is that they may have used nails to avoid the time to make and fit the staircase properly.

TLDR just renailing will work for a short time it is a bodge specially if you don’t use cut nails.

That is certainly one way of doing things, though in my opinion the people who get impressed are those who don’t understand what is involved so I probably don’t want to impress them.

When I do a repair my aim is usually to completely hide that anything has been done. So in this case a simple glue joint is likely sufficient, however if I wanted belt and braces I would add a floating tenon or 2 (still invisible), but of course after the joint has set you can easily drill a few holes for dowels, that they are redundant for the joint strength though of course doesn’t matter as you can use wood for the dowels that doesn’t completely match the stair tread for added emphasis ;) .

As to over engineering it is certainly better than having items fail, so I usually ere on the side of “too much is only just enough”

Creaking of the stairs indicates that they have loosened over time, and that unusual stresses are being given. That is almost certainly why you had a failure. Fix them so they don’t create and they will be OK for another century

did not suggest a repair as that is a little more difficult to do well, however In fact a repair to the broken tread is almost certainly quite easy. There is no need or significant benefit to adding dowels and they add a significant degree of complexity, if the tread has broken as cleanly as it looks a simple glue joint is likely to hold for another 100 years or more, though you will have to make sure the tread makes full contact with the stringers and upright.

There is absolutely no reason to replace treads because of age, wood doesn’t change significantly just because of age. The way the wood has been cut (ie grain direction) is likely the cause of the failure.

However I would carefully inspect the complete staircase to ensure that it is sound, I would fix any treads or risers that show movement.

Remember new does not equal good.

Use the correct anchors and you have no need of all that fafe

i have cabinets hanging on single French cleat battens 30mmx50mm, that hold upwards of 50kg using 3~5 3x3” screws in saw steel anchors

While using epoxy is certainly one way of putting in anchors to aerated concrete it certainly isn’t the only or IMHO the best way.

the best way is to use “saw steel anchors” they are specifically designed for AAC blocks and I have used several hundred of them. My walls are all AAC (aerated concrete)

an mage is at https://sc01.alicdn.com/kf/HTB1QHWYvTlYBeNjSszcq6zwhFXaM/200138850/HTB1QHWYvTlYBeNjSszcq6zwhFXaM.jpg