Which is the most important part of a stereo system?

Well, as often, valuable points of view. I'll go counter-culture. Without excellent power amp nothing will sound right with any source, preamp, speakers or cables. After that turntable itself, not tonearm or cartridge, and speakers - any order. Modest but basically good speakers will sing with great power amp, the other way around won't get you anywhere.

Another vote for the room.
It IS the most important single component affecting the sound by at least half or more of what you will hear.
It's also the most overlooked/disregarded.
The majority of us just keep focusing on the gear.
There's a reason someone came up with "audiofool"...

The room is the most important part. I’ve had a nice setup in a bad room and it sounded really bad. I would rather listen to a mediocre system in a great room over the highest of high end systems in a terrible room any day. The only exception I can think of might be dipole speakers which do a good job of “removing” the room from the equation. 

I think Millercarbon has a strong point. If you change one part of your system it can change everything. If you believe there’s a gap or weak spot that’s probably the best place to start
I recently upgraded my integrated amp - the preamp and amp combo added a DAC snd upgraded my phono stage improving the sound of my streamer, turntable and tape deck. And a friend‘s first comment was ‘I can hear why you like your Sonus Farbers...previous experience was they were analytical and lacking in mid-bass with poor details. Any improvement can improve everything.