Inrush suppresors

So I take it you mean a thermistor, which presents a higher resistance when "cold" and then a much lesser resistance when "warm" after being used for a few minutes... Right?

Obviously, a switch would cause a much faster contact transient, which would be a problem akin to relay arcing, etc. I would avoid this more than I would avoid a thermistor.  A large power transformer can store serious amounts of charge, and you could see inductive kickback, aka an inductive transient, as you switch.  This will eat your switch terminals at best and arc across them at worst.  I've seen it and dealt with it.  It's dangerous and sometimes hard to handle.

Another possibility is that it's a MOV device of some type, unless you're certain it's an NTC thermistor..

First, let me ask you a rhetorical question.. What is the problem you are trying to fix? Is it a purely hypothetical problem?

No transformer should be saturated, that includes power transformers during inrush. Saturation means no more flux fluctuation, akin to clipping a signal. In fact you'd be clipping a 60hz sine wave as well as overheating the core. Power and flux excursions are normal and expected parts of powering up a transformer. A prudent designer has specified enough headroom for the occasional power line fluctuation, and the transformer company has also given a large margin of operation, with the maximum operational VA rating being well below the core saturation point. This is also how they make 50 AND 60hz transformers out of one part, by over-specifying the power rating of the core, to account for the increase in heat during 60Hz usage.

That being said, using a transformer has it's drawbacks, as most everything does. Absolute transient response being one of them. By virtue of isolation, you isolate both the bad and good aspects of readily available line voltage. On the plus side, the lossy and slow nature of flux current transfer also mean that line transients are blunted when transitioning between the primary and secondary sides of the transformer. However, one has to ask, do you deal with transient current demands on the AC or DC side of a design?

Most people answer it on the DC side as it's easier to compensate for. Your DC should be well regulated and well decoupled. The AC side has so many problems with noise and trash (household appliances, lightning, etc) that working on the DC side is much more reliable. If you have transient issues, you need faster (lower ESR) components, lower loss traces, and/or closer decoupling.

Anyway, I'm sure Adcom had it's reasons for adding this part. It could be redundant safety during design, or it could have been added later after finding a problem. The only way to find out is with an oscilloscope and testing.

Funny story about mechanically switching heavy inductive loads.

I was working at a place that had large dc motors on reels of cable. The bosses insisted on using relays and contactors for everything. They didn't trust semiconductors for some strange reason..

But anyway, i was given the task of making a reversible motor controller out of a large variac, rectifier and power contactors (like very large open relays). Against my warnings they insisted (by that i mean they told me to do it or find another job), So i did it.

Every time the motor was as full torque and reversed, an arc would shoot out of the contactor to the chassis and trip the breakers. A good two inch arc!

I eventually built a snubber network to kill as much of the arc as possible but the contactors would wear out after a period of time.

To my chagrin the bosses loved it as they got many billable service orders to fix these things as tree customer was told it was a wearable item.

Uggh.