Possibly Ignorant Power Question

The real measure of an amplifiers ability to drive speakers is current. Current is not a standard measurement, but you can indirectly measure it by measuring the watts at different impedance. Regardless of what speaker you are using you want the one able to produce more current. The closer to watts doubling between 2, 4, 8... the better. I have had massive amps driving speakers not requiring the power and given all things equal the more powerful the better the sound. So for instance my Pass x350 sounded better than a Pass x250.
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There are specially speakers made to be driven by flea powered amps... I assume if you were in that category you would know it.
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Conversely some speakers require lots of power. They don’t do this to be difficult, they have a reason. This is overly simplified. Consider a speaker with a light weight magnet and light material to pull the cone back... it pushes the cone around easily. It also requires little power to do so. But the cone is not stiff and it takes it’s time coming back once extended. So the sound is flabby. Now stiffen the cone and add massive magnets... it requires much more power... but it makes very fast excursions and the cone doesn’t deform while trying to do different sounds at the same time. Much more accurate sound. .
so it is important to find speakers you like and power with at least enough. From my experience if you have a choice of two identical sounding amp the one with greater current grabs the cones better and makes them do what they are supposed to be doing.

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.