Possibly Ignorant Power Question

Good explanation with last answer.  Sufficient current whatever the load (which varies over time with speakers depending on what is being sent to them) is what's important.  So the doubling of watts as ohms halve is a useful rule of thumb that can give you some confidence that the amp will be able to produce sufficient current.  The speaker sensitivity of course plays in as also mentioned, but if looking just at the amp specs, the "doubling" is a nice feature to have at a quick glance.

You ask about 200 watts @ 8, 4, and 2 ohms. That is saying you will have 200 watts available AT 8, 4 or 2 ohms. Nothing less. BUT there is most likely MORE at 4 and 2 ohms..

OP you have to read through the word salad.. BUT That is all that is being said. 

The doubling of wattage is a typology. Nothing more, nothing less.

You will hear class A is all the rage, BUT when you crank most type a amps all the way up, they REALLY start to distort.. Run out of gas, they (Class A) have their own SET of problems :-).  Are they the best?

The same question your asking offers the same answer.
What can you afford?
What do you like?
Are there environmental issues, like HEAT, or kids and animals?

Understand, some amps are just as good SQ wise as others but that may not be what YOU like..

Sound Quality is in the EARS of the beholder.

Toes in the water, AY?

I don't think double the power output at 4Ω  over 8 is much of an indicator of how good an amplifier is... sure it's easier to make one that doesn't double but there's no reason why one that does would sound better.
My advice would be not to go for an amplifier that's more powerful than you actually need. Most if not all amplifiers sound better running at the top half of their rated output so if you go for something four times as powerful as you need then the cost of that unused power is a reduction in sound quality. The other thing is that most volume controls sound worse the more you turn them down as well, with the exception of decent resistor based attenuators. I did a short blog post on amplifier power vs loudness that might be interesting.

Keep in mind that a manufacturer's spec of doubling into 4 ohms is often not accurate when the amp is actually tested.  For example, the specs may say 70wpc into 8 ohms and 140wpc into 4 ohms.  The actual tests could show something like 110wpc into 8 ohms and 150 wpc into 4 ohms.  Technically, the amp exceeds its specs, but it doesn't come close to doubling into 4 ohms.  

"...It should be pretty obvious when using this speaker it is totally irrelevant if amplifier power doubles. All this stuff becomes irrelevant..."
Unless you like speakers like big Magnepans and others.   

"...Try and find the Martin Logan owner who says wow I can use any amp and my speakers sound great! No such thing..."
I'm one. The M-L Electromotion ESL is 91db efficient and I drive them very well with my 25wpc tube amp. Every rule has an exception...I do agree on your efficient speaker premise, it makes life easier for sure.