Is 100-year old armored (BX) cable a safety hazard and should it be replaced?

Agree with Ed Beal's answer about your BX installation likely being safe. Especially so if you don't mess with it and start bending/cracking the brittle old insulation. But answering yr specific question, YES, using AFI breakers will significantly increase the fire safety of any electrical installation, so do it if it gives you peace of mind. Make sure you're attentive to which areas will require a combo GFI/AFI breakers under current code. Adding GFIs will also have the benefit of allowing you to replace 2prong outlets with (suitably labeled) 3 prong. GFIs will sometimes have difficulty holding on old wiring though due to minimal but nonzero leakage current.

I think you are over reacting With what you have provided original would be knob and tube it is still legal. BX is much safer If there is an issue it is inside the flexible conduit and that metal conduit is quite a bit of protection. Replacing breakers may or may not be possible with your existing panel and again your wiring is in conduit so I would not be concerned.

Depending when and Or the type of panel You have we may suggest getting it updated (especially if FPE™, StabLock™, Federal Pacific™ & Zinsco™ are the leading panels that have problems). Some obsolete panels do not have arc fault breakers available but are still good panels.

Note I have seen more problems with skinned NMB than with BX, MC or AC.