What is the order of importance of the parts of an analogue rig?

I'll say the most important is a cart, that you like.

Everything else is a "need to have". To what degree?

I like real heavy EVERYTHING.. Plinths, EXTRA heavy!!

Being clean... Some folks are, some folks aren't...
LPs, take care of them..Handle with care!!!

Vibration control is a BIG thing, for LPs, ay?
It is # 1 right? or close...

Regards

1. Cartridge, but you need to have it well matched to a proper tonearm that won’t cripple its performance. Or else you goofed. But always start at the transducer since they impart the most coloration and your sonic preferences here will be very personal.
2. Phono stage, which needs to be electrically and sonically matched for your chosen cartridge. Includes the SUT if you go that route for MC step-up. Some people can tolerate more or less noise here which will factor into your decision. Choose poorly here, and you’ve also goofed.
3. Turntable needs to be good enough to not destroy the performance capabilities of the above items. This includes necessary isolation from your floor, speakers, other energy sources, etc. Choose poorly here, and you’ve also goofed.

Mainly remember your’re trying to put together a complimentary system of well-matched components. E.g. you can’t go "all in" on a Koetsu cartridge and stick it on a Project Debut. That’s why when it’s time to upgrade, you may find yourself buying all new things, which is why it’s such a money pit :)

Anyways, the way most people buy into their first vinyl system won’t be so orderly. Many will go for the bundle packs, like Project or Rega, with varying degrees of success - though at least the components will be reasonably complimentary, and you just need to choose phono. But if you want to do this "right", you’ll have to spend more than you first think. I started 13 years ago when I fell in love with a consignment SOTA Star III in fiery Koa wood, and just had to have it. It came with a Fidelity Research arm (great arm) and fortunately I didn’t cheap out too bad on the cartridge - got a new Benz Micro Glider (perhaps a slight mismatch to the arm, but not too bad) and a Dynavector P75 phono stage (I could've done a bit better here, but not without spending more). It cost $3.4K all-in, and the setup sounded so good I became a vinyl junkie for life. When I had started looking, I was hoping to keep it down to low-$1K max. Whoops - but I’m so glad I invested much more.

All the above applies. But without a great recorded LP in good condition it’s a waste of money and time.

Being a period press snob, I’m obviously biased.
That’s a separate thread topic altogether.

IMO, may as well have a nice digital setup, if you’re into current music or most of anything post....1980-90? It’s mostly digitally recorded anyhow.

DACs/file playback have come a long way, and CD’s still sound good to me in a proper system. They just don’t have that unexplained magic a good record has.

most significant element depends on the level of analog you are at.

at the entry level your biggest enemy is NOISE, so motor and platter will dominate the musical limitations. cartridge and arm are less significant, phono stage is also less significant.

next level up SPEED STABILITY becomes big and is what gives you the sense of musical flow. so again, motor is huge. better motors separate. they determine whether it sounds real.

next level now we are seeing arms become the limitation. can it relate the BASS and allow the cartridge to sound TRANSPARENT?

then the next level the cartridge and phono stage. how much INFORMATION and TONAL COMPLEXITY.

over about $6k-$7k total..........it’s EVERYTHING.....and at this break point the big separator would be SET-UP PRECISION.

this price point is arbitrary. just about what it takes to get ’over-the-hump’ into better sounding turntables.....from more modest level turntables. and if you go with the right vintage set-up and you know your stuff you could exceed this performance level for less money.

just how things look to me from being a serious analog guy for many years.

i have three separate $75k-$100k turntable systems. there are no pieces that you could significantly lower the performance and get the same result. they are SYNERGISTIC SYSTEMS.

you could make the case that over about $40k-$50k that the turntable/motor/plinth are most significant....but ONLY because they are the most expensive pieces. i'm not sure they are most important to the performance.

one more very significant thing.

past a certain point......say over $15k-$20k mostly any turntable is good enough that the specific pressing matters more than the turntable.

the media is a bigger issue than the gear once the gear gets to a certain level.

which is why many vinyl lovers chase pressings and not gear. a better use of time and resources.