I suspect the reason is to provide a dedicated circuit for an iron (yeah, yeah, don't get me started on people ironing in places other than the laundry).
The main reason for the dedicated bathroom circuits is for hair dryers (if you're living in an older house with teenage girls you know all about that one).

What's not defined is the "extent" of the laundry area or how many receptacles you can install in that "area."

"Any idea what the reasoning behind the code that you can't use any circuit in the Laundry room to anywhere else?"
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Simple:

Washers are motorized appliances. Motorized appliances required a large 'jolt' of power to get them started, and therefore should not have to compete for power with other loads on any circuit, otherwise you risk nuisance tripping of breakers at best, or overheating of conductors and the fire at worst.

To avoid overloading, conductors and overcurrent protection devices for motorized appliances (like laundry circuits and refigerator circuits) need to be sized for 125% of the anticipated load...and then sized for another 125% if the loads will be run continuously (IE: 3 hours or more continuously).

This generally means a 'dedicated' circuit/receptacle for the washer alone.

With other power and lighting loads in the laundry area on separate circuits.

NEC 210.11c2 code does not say the washer needs to be on a dedicated circuit. It allows all the circuits in the laundry room to be shared on a single circuit and does not specify how many - so if you put 8 receptacles in the laundry and had that shared with the washer that is fine. You just can't also share it with anything in any other nearby room/area like the garage.

In my situation I happen have 3 circuits going to the laundry "area." One for the washer, one for lights & a cpl plugs. And a 3rd circuit for a gas dryer and a cpl other plugs even though this is not required.

Because those 3 circuits "all" are in the laundry area - I can't share any of them with anything outside of the laundry area - by my reading of the code. It seems a waste since I could really use one of those circuits for the garage plugs.

If anything, it is in my garage where I have table saws and chops saws which sometimes resets the GFCI receptacle when I turn them on anywhere in the garage. There is only one 20 amp circuit for all the garage plugs. It does not share with anything else like garage lights/door opener/furnace, but the GFCI still resets every other week when I initially run the saws. Then I reset the GFCI, and it works fine for a while. I have changed the GFCI twice with no difference.

Kisu has this one right. The receptacles in the laundry area must be served by a 20A circuit or circuits and not shared with anything else. As pointed out earlier it has NOTHING to do with the washing machine specifically. If they wanted it to apply to a specific appliance they would have said so.

This might sound like a philosophical question, but since there are no walls here, is this a laundry 'area'? If the gas dryer were on the same circuit as the washer, could that old gas circuit be used for something else since no laundry items would be connected to it?

As I stated, the term "laundry area" is nebulous. If you had a dedicated laundry room, that would be pretty obvious. If you have the washer and dryer int he garage, how far does the "area" extend from the washer? That would be up to your AHJ.

The "no other outlets" means you can use that circuit to feed lighting or receptacles located outside the area. It doesn't mean you can't plug non-laundry devices in there. My laundry was in the back hallway and I always used to plug my power saw in there when I was working in the garage as the single garage receptacle was only a 15A.

And if there is enough abuse of the 'laundry circuit' you can look forward to tighter definition.

Especially if someone can figure a way to make money from the tighter definition.