Do you belong more to souce first or to speakers first school of thought ?

The speakers are what I'm actually hearing, so I'd possibly be speakers first, but if the source of what the speakers are playing is faulty. "Garbage In, Garbage Out", or 'GIGO' for short, btw, is an old, old expression - used it with IT along with other systems that take something in and spit something else out, for many many decades. Nobody at Linn invented it. 

If you are talking about creating a system and not planning on gradual, incremental improvements, the answer might be different than someone establishing a strategy to improve the system.  Linestages, followed by digital source components, are MUCH more consistent in their sound than speakers and power amps.  The difference between extremely good versions of these components and more price competitive models is relatively small as compared to differences in amps and speakers.  So, it might make sense to invest a bit heavily in those components with an expectation that they will be good in any system and will be long-term keepers.  This would be a strategy for someone planning to make incremental improvements, whereas someone buying a system that will not be changing much would probably be better off spending more on the amp and speakers.

Speakers and amps and the room housing the speakers should be considered somewhat together.  There are very big differences in the sound of speakers, and they may have extremely different demands as to room placement and compatibility.  Because of such variability in sound, one will tend to make mistakes on choices early on before becoming more certain of one's own preferences and what might work in one's setting.  That might argue for not spending so much on speakers early on in the process.  The amp should at least roughly match they type of speaker it will be asked to drive in terms of power output and compatibility with the impedance curve of the speaker.  For example, a tube amp with high output impedance might not match well with a speaker with a low impedance that also has other related demands.

I personally find certain types of amplifiers to be significantly better sounding than other types (I like low-powered tube amps), so any speaker candidates will have to be compatible with that category of amps.  It is MUCH more important to prioritize the right choice of amp than it is to prioritize choice of sources or the linestage-- good choices there will almost always be good with whatever choice is made for amp and speaker.  

To me, the source is somewhat discrete relative to the rest of the system. A crappy source will sound crappy with whatever follows. However, the speaker/amplifier relationship is much more symbiotic, as several here have said. I believe somebody mentioned matching the speaker and the room but I haven’t seen anyone yet mention the relationship between the type of music listened to and the speaker. Sure a good speaker should sound good with most music but not every speaker can handle the power to fill a large room to believable levels with hard rock music, as one example. Also, speakers can alter tone as not every speaker projects a flat frequency response so some offer a presentation that seems to add a bit of warmth while others can be perceived as providing greater detail. In addition, loud is not enough for some types of music or for some listeners where there needs to be a feeling of weight and power behind the sound, as it is in real life. Conversely, not every speaker sounds good, or realistic, at low volumes. Having subs, or not, can also make a big difference.

All that said, I would start by finding the right speakers for my room, music, and listening preferences, then match an amplifier that can drive them convincingly, and finally work on improving source components. I view the preamp as sort of between the amplification/speaker side of things and the source, as in my experience the right preamp for a system can make it sound wonderful while the wrong preamp can be similar to ruining your favorite dish by adding the wrong spices or flavorings.

Speakers + room for me. I have a loft that is medium size but it opens to a stairwell on the left side. That means the left speaker is separated by the stairwell with a half wall while the right is adjacent to a typical 9' high wall. Not all speakers like this configuration but it's what I have. Given that my open baffle speakers really shine in this room with only minimal corner treatment. 

In summary: 1) sort out the room for the best listening position, 2) choose speakers that work well in the room, 3) adjust any treatment to account for room issues, 4) build the front-end that makes the whole system work. 

Personally I really don’t belong to either / any, because it all matters. The generality of the speakers matter most is a good rule of thumb for the folks fairly new to pursuit of the high end… but it holds best when one has appropriate portioned investments in system components with traditional speaker technology. But can be wrong with specialty speaker technology (ie, planar) unique individual components of systems.