Not always.
It is very possible for the inrush to fry the MOS FETs. And in many cases, when they fail, they can fail in a short circuit. The BMS is a small computer that is running instructions. It has to measure the overload current and then react to it and then try to turn off the FETs. Each instruction takes time. Sure, a fast processor might do it in just a few milliseconds, but the FETs can be toast in a microsecond at 5,000 amps. They don't use super fast processors because they don't want the BMS using any more power than it has to to do it's job.
And also yes.. The huge inrush currents from a lithium battery bank can damage the filter capacitors as well. Even with lead acid batteries, most inverters recommend using a pre-charge resistor to avoid damage from the high inrush surge. It is also a good idea for charge controllers as well. When I first connected my Victron charge controller, it made a pretty healthy spark. Every time since, I have been using my resistor to be nice to it. It certainly didn't pull near the inrush that the inverter does, but it was enough that I feel the pre-charge is a good idea.