Which is the most important part of a stereo system?

Too hard to answer. Every component is critically important. Without one, you have nothing.

Here is my prospective on what's the most important part of a great sounding hifi system.
You can easily find excellent source, or amplifier, or speakers or cables.  However, there are not so many good sounding preamplifiers and they make enormous difference for the quality of sound, even if you only use digital source. 

OIL first! oh wait... wrong forum. ;-)

@shangyien said "The weakest part of your system is the most important."

My thoughts exactly.

With the arrival of digital sources many have now come to accept that transducers (loudspeakers, cartridges, arms, turntables and microphones) are where you will find the bottlenecks in modern audio playback systems. (Room acoustics and source mastering are also important but perhaps separate issues here.)

In much the same way that traditional rotating hard drives are now accepted as generally the major bottleneck (ahead of both RAM and processor speed) in PCs.

Differences in all digital playback sound (or even in amplifiers) are barely measurable let alone readily identifiable in listening tests.

On the other hand the sonic differences between loudspeakers and turntables (arms/cartridges) are difficult to deny.

So having said that, your idea of upgrading your Pro-Ject to something like a Technics 1210 makes great sense.

Upgrading your B&W speakers on the other hand is going to be more of a challenge as far as I can see. Especially if you want across the board improvements including both dynamics and bandwidth.

It can be done and if successful, it will be worthwhile in simply that everything played back through them will sound better.

Maybe that could be a project for a future day? Certainly plenty, almost infinite number of candidates out there including various Harbeths, PMCs, Tannoys, JBLs, ATCs, Wilson’s, Zu’s, larger B&W’s etc

All with very little consensus as to what’s best.

I order:  HD source(or turntable), then speakers, then amp, the rest .  With that being said, the speakers change the sound the most and it is a matter of taste.