Turntable versus tonearm versus cartridge: which is MOST important?

Right. The comparisons have already been done. I’ve heard it myself, a couple times now. Moving the same Benz Ruby from a Graham arm to Origin Live Conqueror was a vastly bigger improvement than moving from the Glider to the Ruby. Have also moved the same arm/cartridge from one table to the next. Another huge difference.

What you need to keep in mind is any cartridge however good or expensive it may be, all it can do is wiggle back and forth and up and down. All that happens is the really good expensive ones let you hear more of the tiniest details as it wiggles around. But the cartridge itself, no matter how good, has no way of knowing what wiggles are signal coming from the groove and what wiggles are vibrations coming from the turntable and arm. It dutifully sends it all right along to the phono stage, the obscuring noise just as much as the music signal. 

Turntable and arm on the other hand, when they are the very best they produce vanishingly small detail-smearing vibrations. Because of this even a very inexpensive cartridge will play back so much more of what is in the groove you can hardly believe it. Whereas if you take a really expensive cartridge and mount it on a cheap arm so much of what you are hearing is noise from the arm and table you will wonder why you spent all that money in the first place. What a waste. It is even entirely possible you get to such a resolving cartridge and phono stage as to let you hear all the individual faults of the arm and table!

In other words your hunch is a bad one. The truth is closer to the opposite. Put your money into table and arm. Then cartridge.

a good $500 cartridge on a $10K table/arm?

I would start from there

A state of the art $10K cartridge on a $500 table/arm or a good $500 cartridge on a $10K table/arm?

Now that I think about it, I actually have everything here to make almost this exact comparison: Origin Live Sovereign turntable with Origin Live Enterprise arm is very close to $10k. Koetsu Black Goldline is not a $10k cartridge but is a pretty darn good cartridge all the same.   

My vintage Technics SL1700 is also very close to your hypothetical $500 turntable. On it is a Stanton 681EEE which is quite a bit less than $500.   

So not exactly what you said, but pretty darn close. In this case it is no contest. I am not about to move the Koetsu to the Technics, not unless you get off your hypothetical you know what and make the trip here. But if you do and if we do I will bet you your next paycheck against mine you agree the table and arm make the biggest difference. 

And I hope you have a lot of OT on your next one because I sure have a lot of OT on mine!

Yeah. What they said.

I remember a demo I sat in on back in the 80s. Linn LP12 with a K9 (cheap MM AT cartridge) on an Ittok beat a Karma (MC) on a cheap Basik arm.

So, which would sound better:

A state of the art $10K cartridge on a $500 table/arm or a good $500 cartridge on a $10K table/arm?

It’s a weird question, you’re trying to put together something very cheap and something very expensive.

Ask yourself who can afford a $10k cartridge and why ? A person who wishes to buy a $10k cartridge for a $500 turntable must be sick!

Now forget about retail prices, because on the secondhand market you will find something expensive for cheap (used), the price drop can be 50% (sometimes even more). If you will stay away from some popular brands like restored Garrard (for example) you can actually find some very nice turntables for a reasonable price (Denon DP-80 Direct Drive for example), same for vintage tonearms in perfect condition.

If we already have a very expensive turntable (new or vintage) with a nice tonearm (new or vintage) we can always find NEW or VINTAGE cartridges for that arm. A good vintage high-end cartridge (NOS sample) can put your overpriced new ultra high-end to the dust forever, this is where the price difference can be shockingly big, but a price/performance ratio is not that big at all ! It will break your concept, because your NEW $10k LOMC is overpriced while a $1k VINTAGE is underrated and not supported by manufacturers ads anymore (can be totally unknown to most of the audiophiles today).

A question like this makes no sense at all, because these numbers do not mean anything, except for a situation where you are standing in the audio shop in front of the new components and for some reason pretend that a new $500 turntable that look like a toy for your children is better with a $10k cartridge (or vise versa).

In the real world you can buy amazing turntables for under $3k and amazing high-end cartridges under $3k estimate (you can do it even for a half price:). But well matched components (tonearm/cartridge) are better.